The organization of this thesis is as follows : The thesis begins in Chapter 1 that contains a brief history about the appearance of the concept of fractional derivation and a description of some physical problems that are modeled by fractional differential equations. In the sequel, we describe some basic concepts and give some information about direct and inverse problems. Afterward, we collect the results obtained from four articles above and present them respectively in Chapters 2, 3, 4 and 5.

The next three chapters are structured on Inverse Problems as follows : We start in Chapter 2, by studying an inverse problem in fractional calculus. Using the Fourier method, we prove two theorems of existence and uniqueness for the solutions of fractional order differential equations with involution. In Chapter 3, we consider a class of inverse problems for restoring the forcing term of a fractional heat equation with involution and present the results on existence and uniqueness of solutions of these problems. We also discuss delay fractional order differential equaix tions to achieve some related results. In Chapter 4, we study two inverse problems concerning the wave equation with a perturbative term of involution type with respect to the space variable. We obtain existence and uniqueness results for these problems based on the Fourier method.

The last part is devoted to studying a nonlinear sequential differential equation of Blasius type with Caputo fractional derivative. We reduce the problem to the equivalent nonlinear integral equation and prove the complete continuity of the nonlinear integral operator. We prove also the existence of a solution of the problem for the Blasius equation of fractional order. We present it in Chapter 5.

acknowledgement

W ith the support of the people with whom I have worked and met throughout my research this thesis has become the key to my rewarding life.

First and foremost, I thank my advisor Prof. Mokhtar Kirane who encouraged me to work in the field of Fractional Calculus and it's application to Inverse Problems. It was a great pleasure completing this work under his guidance. I am deeply indebted to him for his constant guidance and encouragement. I am grateful for his collaboration, thorough discussions, and reviews of my work, as well as for pushing me forward whenever I needed it. With his great knowledge and open attitude, he created a fertile environment of which this thesis is the yield. I deeply thank my two reporters Prof. Etibar Penahli and Prof. Mohammad Guedda for having spent time reading my manuscript. I would also like to thank the members of the jury. I am honored by the presence of professors Manuela Rodrigues, Abdallah El Hamidi and Laurence Cherfils in my defense to judge this thesis. Some of them come from far away and all of them certainly have a lot to do through their teaching and research activities. I warmly thank them for having accepted with great sympathy to be a member of my jury.

I also thank Isabelle Hirsch, Jennifer De La Corte Gomez, Nadine Seguin and Camille Moinard, secretaries of the University of La Rochelle, for their assistance with administrative procedures. I want to thank my colleagues for creating a pleasant work environment. I really appreciate all the good moments that we shared together. Special thanks to Dr. Berikbol Torebek for his contributions to the works we have carried out, and to my former colleague Dr. Rafika Lassoued for her critical analyzes. The discussions I did with them allowed me to establish very constructive exchanges during my Ph.D. studies. I extend my sincere thanks to my friends from all over the world, especially to Fatih Cosgun, Mevlut Bicici and his lovely family, Muzzamil Luqman and Ugur Pinar who supported me spiritually and morally during this journey.

Finally, my heartfelt thanks to my parents for supporting my deciv sion of opting for an academic-research career. Without their blessings this thesis could not have been conceived. They have been the tower of strength and motivation for me. Their devotion for my betterment will always be reflected in every milestone I achieve in my life.

Résumé

Cette thèse est le résultat de mes recherches durant mon doctorat à l'Université de La Rochelle. La plupart du matériel de la thèse est basée sur les quatre articles suivants publiés durant cette période : L'organisation de cette thèse est la suivante : La thèse commence par le Chapitre 1 qui contient un bref historique sur l'apparition du concept de dérivation fractionnaire et un description de quelques problèmes physiques modélisés par des équations différentielles fractionnaires. Puis, nous décrirons quelques concepts de base et donnons quelques informations nécessaires sur les problèmes directs et inverses. Ensuite, nous recueillons les résultats obtenus à partir de quatre articles ci-dessus et les présentons respectivement dans les chapitres 2, 3, 4 et 5.

Les trois chapitres suivants sont structurés sur les Problèmes Inverses comme suit : Nous commençons le Chapitre 2, par l'étude d'un problème inverse à dérivée fractionnaire. En utilisant la méthode de Fourier, nous prouvons deux théorèmes sur l'existence et l'unicité de solutions d'équations différentielles d'ordre fractionnaire avec involution. Dans le Chapitre 3, nous considérons une classe de problèmes inverses pour restaurer le terme forcing d'une équation fractionnelle de chaleur avec involution et présentons les résultats sur l'existence et l'unicité des solutions pour certaines valeurs de la condition initiale de ces problèmes. Nous discu-tons également des équations à retards aux dérivées fractionnaires afin d'obtenir certains résultats connexes. Dans le Chapitre 4, nous étudions deux problèmes inverses concernant l'équation d'onde avec un terme perturbatif de type involution par rapport à la variable d'espace. Nous obtenons des résultats d'existence et d'unicité pour ces problèmes, basés sur la méthode de Fourier.

La dernière partie est consacrée à l'étude d'une équation de type Blasius différentielle séquentielle non linéaire avec une dérivée de Caputo. Nous réduisons le problème à une équation intégrale non linéaire et prouvons ensuite la continuité complète de l'opérateur intégral non linéaire. Nous démontrons l'existence d'une solution du problème pour l'équation de Il a fait un développement rapide et s'est révélé comme un outil puissant dans la modélisation de certains phénomènes dans plusieurs domaines de la sciences tels que la physique, la chimie, la biologie, l'ingénierie et la finance, surtout au cours des trois dernières décennies. Les dérivées fractionnaires et les intégrales fractionnaires représentent un comportement non local (en raison de l'intégrale impliquée dans la définition) associé aux effets de mémoire. Calculer une dérivée fractionnaire à un certain moment exige tous les processus précédents avec des propriétés de mémoire [START_REF] Podlubny | Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications[END_REF]. C'est l'avantage principale du calcul fractionnaire d'expliquer les processus associés aux systèmes physiques complexes qui ont une mémoire à long terme et / ou des interactions spatiales à longue distance. De plus, les équations différentielles fractionnaires peuvent nous aider à réduire les erreurs découlant de paramètres négligés dans la modélisation des phénomènes physiques [START_REF] Arafa | Stability analysis of fractional order hiv infection of CD4+ T cells with numerical solutions[END_REF], [START_REF] Mainardi | Fractional calculus and waves in linear viscoelasticity: an introduction to mathematical models[END_REF]. Ce thème de recherche peut mieux traduire la réalité de la nature des phénomènes de la vie réelle, ce qui l'a rendu plus populaire au sein de la communauté des chercheurs et des ingénieurs [START_REF] Machado | Some applications of fractional calculus in engineering[END_REF].

Les champs d'application du calcul fractionnaire s'étendent rapidement. En génie mécanique, il existe plusieurs applications du calcul fractionnaire, par exemple, l'étude des systèmes de contrôle [START_REF] Matouk | Dynamical behaviors, linear feedback control and synchronization of the fractional order Liu system[END_REF], modélisation de la diffusion anormale [START_REF] Wei | A coupled method for inverse source problem of spatial fractional anomalous diffusion equations[END_REF]. Le comportement non-standard ou anomalie de l'équation de réaction-diffusion, équation de transport, est connu sous le nom de diffusion anormale non gaussienne, qui a des effets de mémoires longues. Une solution de modèle de diffusion anormale basée sur le calcul fractionnaire à différentes conditions d'opérations permet d'obtenir de meilleurs résultats [START_REF] Metzler | The random walk's guide to anomalous diffusion: a fractional dynamics approach[END_REF]. En physique, il existe plusieurs applications potentielles de dérivées fractionnaires. Une monographie intéressante sur les applications des intégrales et des dérivées fractionnaires à la physique des polymères, à la biophysique, à la thermodynamique, à la rhéologie et aux systèmes chaotiques a été éditée par R. Hilfer [START_REF] Hilfer | Fractional time evolution[END_REF]. De plus, en médecine, il a été déduit que les membranes des cellules d'un organisme biologique ont une conductance électrique d'ordre fractionnaire et ensuite, elles sont classées dans des groupes de modèles d'ordre non-entier. Les dérivées fractionnaires incarnent les caractéristiques essentielles du comportement rhéologique cellulaire. Ils ont un grand succès dans le domaine de la rhéologie [START_REF] Arafa | Stability analysis of fractional order hiv infection of CD4+ T cells with numerical solutions[END_REF].

Actuellement dans la littérature mathématique, il existe plusieurs définitions des dérivées fractionnaires. Parmi les plus populaires, citons la dérivée de Grünwald-Letnikov, Riemann-Liouville, Caputo et Riesz-Feller [START_REF] Podlubny | Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications[END_REF]. Même si elles sont différentes, elles sont toutes liées les unes aux autres. Grünwald-Letnikov est l'approche la plus évidente pour définir les dérivés fractionnaires. Elle est principalement utilisée pour l'approximation numérique des dérivées fractionnaires. La derivée de Riemann-Liouville a joué un rôle important dans son application en mathématiques pures tandis que Caputo a été mise en place pour répondre aux problèmes appliqués. Caputo a été le premier à appliquer le calcul fractionnaire à la mécanique, en particulier aux modèles linéaires de viscoélasticité [START_REF] Caputo | Linear models of dissipation whose Q is almost frequency independent-II[END_REF], [START_REF] Caputo | Elasticitá e dissipazione (Elasticity and anelastic dissipation)[END_REF]. Les dérivées de Caputo permettent l'utilisation des conditions initiales physiquement interprétables, ce qui n'est pas autorisé par la derivée de Riemann-Liouville. Un bref aperçu historique du développement du calcul fractionnaire est donné par Ross [START_REF] Ross | A brief history and exposition of the fundamental theory of fractional calculus[END_REF]. Le manuel de Oldham et Spanier [START_REF] Oldham | The fractional calculus theory and applications of differentiation and integration to arbitrary order[END_REF] est concerné par les définitions et les propriétés des opérateurs intégro-différentielles d'ordre fractionnaire. En 1987, un livre encyclopédique a été écrit par Samko, Kilbas, et Marichev [START_REF] Samko | Fractional integrals and derivatives: theory and applications[END_REF]. Une présentation des nombreuses applications issues du calcul fractionnaire est présentée dans Podlubny [START_REF] Podlubny | Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications[END_REF]. Récemment, plusieurs mathématiciens et chercheurs ont obtenu des résultats et des généralisations importantes de la modélisation des processus réels à l'aide de calcul fractionnaire ( [START_REF] Arafa | Stability analysis of fractional order hiv infection of CD4+ T cells with numerical solutions[END_REF], [START_REF] Debnath | Recent applications of fractional calculus to science and engineering[END_REF], [START_REF] Ferreira | Fundamental solution of the multi-dimensional time fractional telegraph equation[END_REF], [START_REF] Ortigueira | What is a fractional derivative[END_REF], [START_REF] Soubhia | Theorem for series in three-parameter Mittag-Leffler function[END_REF]).

Revue historique sur les Problèmes Inverses

En science, un problème inverse est une situation dans laquelle à partir d'observations expérimentales, on cherche à déterminer les causes d'un phénomène. La théorie mathématique des problèmes inverses a été essentiellement ignorée jusqu'au milieu du vingtième siècle. Au lieu de cela, les scientifiques se sont concentrés sur des problèmes directs, c'est-à-dire la construction du modèle lui-même plutôt que le processus d'inversion. Puisque le modèle lui-même est inexact, un tel processus d'inversion entraîne généralement des problèmes d'existence et de stabilité. Au début du vingtième siècle, l'idée de problèmes directs dominait la physique mathématique. En effet, le mathématicien français Hadamard estimait qu'un problème physique important devait être bien posé, c'està-dire que le problème devait toujours avoir une solution unique qui dépend continûment des données. Cette idée a persisté au milieu du vingtième siècle. Cependant, l'avènement de la mécanique quantique et de nombreux problèmes dans les domaines de la physique classique tels que la conduction thermique et la géophysique ont lentement convaincu les mathématiciens et les scientifiques que les problèmes directs n'étaient pas les seuls problèmes scientifiques et que la théorie mathématique des problèmes inverses commençait à être développé par des mathématiciens de l'Union Soviétique dirigés par Tikhonov. La solution d'un problème inverse consiste à inverser le modèle pour récupérer des informations cachées sur les phénomènes physiques à partir des observations. Une étude complète de nombreux domaines des problèmes inverses et de l'imagerie peut être trouvée dans [START_REF] Scherzer | Handbook of mathematical methods in imaging[END_REF]. Les premières publications sur les problèmes inverses et les problèmes mal posés remontent à la première moitié du XXe siècle. Leurs sujets étaient liés à la physique, la géophysique, l'astronomie et d'autres domaines de la science. Depuis l'avènement des ordinateurs puissants, le domaine d'application de la théorie des problèmes inverses et des problèmes mal posés s'est étendu à presque tous les domaines de la science qui utilisent des méthodes mathématiques. La résolution de problèmes inverses peut également aider à déterminer la localisation, la forme et la structure des intrusions, des défauts, des sources (de chaleur, d'ondes, de différence de potentiel, de pollution), etc. Compte tenu d'une telle variété d'applications, il n'est pas surprenant que la théorie des problèmes inverses et des problèmes mal posés soit devenue l'un des domaines de la science moderne qui se développe le plus rapidement depuis son apparition.

Problèmes Directs

Pour définir diverses classes de problèmes inverses, nous devons d'abord définir le problème direct. En effet, quelque chose "inverse" doit être le contraire de quelque chose de "direct". En général, les problèmes directs sont bien posés [START_REF] Kirsch | An introduction to the mathematical theory of inverse problems[END_REF]. Le concept mathématique de problème bien posé a été proposé par Hadamard en 1932. Il croyait que les modèles mathématiques des phénomènes physiques devraient avoir les propriétés suivantes:

•Une solution existe;

•La solution est unique;

•Elle dépend continûment de la donnée. La première condition décrit la cohérence du modèle mathématique, la deuxième reflète la précision de la situation réelle et la troisième condition exprime la stabilité de l'équation, c'est-à-dire qu'un petit changement dans l'équation ou dans les conditions latérales entraîne un léger changement de la solution. Un problème qui ne satisfait pas à l'une des conditions précédentes est un problème mal posé.

En physique mathématique, un problème direct est généralement un problème de modélisation de certains champs, processus ou phénomènes physiques (acoustique, électromagnétique, chaleur sismique, etc.). Le but de résoudre un problème direct est de trouver une fonction qui décrit un champ ou un processus physique en n'importe quel point d'un domaine donné à tout instant (si le champ est non stationnaire).

Les problèmes directs pour les équations de diffusion fractionnaire telles que les problèmes à valeur initiale ou limite ont été étudiés en détail dans ( [START_REF] Eidelman | Cauchy problem for fractional diffusion equations[END_REF], [START_REF] Luchko | Some uniqueness and existence results for the initialboundary-value problems for the generalized time-fractional diffusion equation[END_REF], [START_REF] Metzler | Boundary value problems for fractional diffusion equations[END_REF], [START_REF] Sakamoto | Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems[END_REF]) et les références qui y figurent.

Problèmes Inverses

Un problème inverse est généralement mal posé. Un problème mal posé est un problème qui ne répond pas à l'un des trois critères de Hadamard pour être bien posé, c'est-à-dire, des petits changements dans les données de mesure entraînent des changements indéfiniment importants dans la solution. La plupart des difficultés à résoudre des problèmes mal posés sont causées par l'instabilité de la solution. Par conséquent, l'expression "problème mal posé" est souvent utilisée pour des problèmes instables. A cette époque, on pensait que les problèmes naturels devaient avoir des solutions mathématiques continues; on pensait que cela faisait partie de l'ordre inhérent des choses. Depuis lors, nous avons découvert que de nombreux problèmes scientifiques et techniques importants ne sont pas, en fait, bien posés au sens traditionnel, car ils n'ont pas des solutions continues. Les problèmes inverses et mal posés ont commencé à être étudiés et appliqués systématiquement pour fournir des informations à de nombreuses applications dans différents domaines. Cela inclut les problèmes en médecine (par exemple, dans les tissus organiques en imagerie médicale par résonance magnétique), en physique (mécanique quantique, acoustique, etc.), en économie (en théorie du contrôle optimal, etc.) et tous les autres domaines où les méthodes mathématiques sont utilisées (voir, par exemple, [START_REF] Bertero | Introduction to inverse problems in imaging[END_REF], [START_REF] Epstein | Introduction to the mathematics of medical imaging[END_REF], [START_REF] Uhlmann | Inside out: inverse problems and applications[END_REF] et [START_REF] Uhlmann | Inverse Problems and Applications: Inside Out II[END_REF]). L'un des premiers problèmes inverses résolus dans le passé était la découverte de Newton des forces qui font que les planètes se déplacent conformément aux lois de Kepler. Des recherches sur la structure interne de la croûte terrestre ont impliqué des champs électromagnétiques dans la théorie des problèmes inverses.

Tichonov [1963] fut le premier à traiter des problèmes mal posés, introduisant ainsi le concept de régularisation. Un problème mal posé devra souvent être régularisé ou reformulé avant de pouvoir procéder à une analyse numérique complète à l'aide d'algorithmes numériques. La régularisation demande souvent de nouvelles hypothèses pour affiner complètement le problème et le réduire. L'idée de la méthode de régularisation est de remplacer le problème mal posé par un problème bien posé, ce qui peut être fait en introduisant un opérateur régularisé qui considère les informations préalables concernant la solution exacte. Dans le domaine des problèmes inverses non linéaires, il y a beaucoup plus de progrès à faire. D'autre part, les équations différentielles fractionnaires deviennent un outil important dans la modélisation de nombreux problèmes de la vie réelle et il y a eu donc un intérêt croissant pour l'étude des problèmes inverses avec des équations différentielles fractionnaires ( [START_REF] Aleroev | Determination of a source term for a time fractional diffusion equation with an integral type overdetermining condition[END_REF], [START_REF] Furati | An inverse problem for a generalized fractional diffusion[END_REF], [START_REF] Kirane | An inverse source problem for a two dimensional time fractional diffusion equation with nonlocal boundary conditions[END_REF], [START_REF] Zhang | Inverse source problem for a fractional diffusion equation[END_REF]). Les premiers résultats mathématiques pour le problème inverse de trouver un coefficient de diffusion pour une équation différentielle fractionnaire sont obtenus dans [START_REF] Moore | On the uniform convergence of the developments in Bessel functions[END_REF]. De nombreux types de problèmes aux limites, y compris les problèmes directs et inverses, ont été formulés pour les différents types d'EDP d'ordre entier et avec plusieurs opérateurs différentiels d'ordre fractionnaire. Il existe de nombreux travaux sur l'étude des problèmes directs et inverses pour les équations de diffusion fractionnaire en temps ou d'onde avec la dérivée de Caputo (voir [START_REF] Feng | Inverse source problems for time-fractional mixed parabolic-hyperbolic-type equations[END_REF], [START_REF] Luchko | Some uniqueness and existence results for the initialboundary-value problems for the generalized time-fractional diffusion equation[END_REF], [START_REF] Sakamoto | Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems[END_REF]).

Dans cette thèse, nous nous intéressons à l'existence et l'unicité des solutions de problèmes inverses pour les équations différentielles fractionnaires en temps.

Introduction (English) 1.2.1 Fractional Calculus

The Non-Integer Order Calculus, traditionally known as Fractional Calculus is the branch of mathematics that tries to interpolate the classical derivatives and integrals and generalizes them for any orders, not necessarily integer order. But with this definition, many interesting questions will arise; for example, if the first derivative of a function gives you the slope of the function, what is the geometrical meaning of half derivative? In half order, which operator must be used twice to obtain the first derivative? The early history of this questions goes back to the birth of fractional calculus first appeared in the correspondence of Leibniz with L'Hospital (1695), Johann Bernoulli (1695), and John Wallis (1697) as a mere question or maybe even play of thoughts. Many mathematicians focused on this topic. However, nothing much has been done in the field. One of the reasons is that the mathematical tools of fractional calculus were not available. Another reason is the lack of practical applications of this field. Nevertheless, beginning with the nineteenth century, interesting developments have been made in the theory of Fractional Calculus: Laplace (1812), Lacroix (1812), Fourier (1822), Abel (1823-1826), Liouville (1832-1837), Riemann (1847), Grünwald (1867-1872), Letnikov (1868-1872), Sonin (1869), Laurent (1884), Heaviside (1892-1912), Weyl (1917), Davis (1936), Erdèlyi (1939Erdèlyi ( -1965)), Gelfand andShilov (1959-1964), Caputo (1969), and many others. Yet, it is only after the first specialized conference organized by B. Ross on "Fractional Calculus and its applications", the fractional calculus has become one of the most intensively developing areas of mathematical analysis. Many applications of various kind of fractional differential equations became a target of specialists due to both theoretical and practical reasons [START_REF] Baleanu | New trends in nanotechnology and fractional calculus applications[END_REF]. It has gone through a rapid development and has been revealed as a powerful tool in the modeling of certain phenomena in several sciences as Physics, Chemistry, Biology, Engineering, and Finance especially during the past three decades.

Considering a differential equation that describes a specific phenomenon, a common way to use fractional modeling is to replace the integer order derivatives by non-integer derivatives, usually with order lower than or equal to the order of the original derivatives, so that the usual solution may be recovered as a particular case [START_REF] Podlubny | Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications[END_REF].

A simple example of fractional derivatives of the function f (t) = t 2 , is plotted for different values of the fractional order in Figure 1. The different values of the fractional order are obtained using the expression

d m dt m t p = Γ (p + 1) Γ (p + 1 -m) t p-m ,
where p is a real number. Actually, when modeling real physical phenomena, fractional derivatives can provide more accurate results than integer order derivatives dt m f (t), of a quadratic function, f (t) = t 2 (blue, solid line) with the order m which has values 0.25 (magenta, dotted line), 0.50 (black, dashed line), 0.75 (green, dash-dot line), and 1 (red, thicker solid line). [START_REF] Machado | Some applications of fractional calculus in engineering[END_REF]. The advantages of fractional derivatives are that they have a greater degree of flexibility in the model and provide an excellent instrument for the description of the reality. This is because of the fact that the realistic modeling of a physical phenomenon does not depend only on the instant time, but also on the history of the previous time, i.e., calculating timefractional derivative at some time requires all the previous processes with memory and hereditary properties [START_REF] Podlubny | Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications[END_REF]. It exists also in many biological systems ( [START_REF] Diethelm | Algorithms for the fractional calculus: a selection of numerical methods[END_REF], [START_REF] El-Sayed | On the solutions of time-fractional bacterial chemotaxis in a diffusion gradient chamber[END_REF]).

Fractional derivatives and integer order derivatives are both linear operators. However fractional derivatives are usually nonlocal operators while integer order derivatives are local operators. As shown in Figure 2, the integer order derivative of a function at a point depends only on the local behavior of the function. However the value of the fractional derivative at a point depends on the entire behavior of the function [START_REF] Schumer | Fractional advection-dispersion equations for modeling transport at the Earth surface[END_REF].

The application areas of fractional calculus is expanding rapidly. The increasing interest in fractional differential equations are motivated not only by their application to problems from viscoelasticity, heat conduction in materials with memory, electrodynamics with memory, and also because of they can be employed to approach nonlinear conservation laws. Besides, fractional differential equations can help us to reduce the errors arising from the neglected parameters in modeling real-life phenomena [START_REF] Arafa | Stability analysis of fractional order hiv infection of CD4+ T cells with numerical solutions[END_REF], [START_REF] Mainardi | Fractional calculus and waves in linear viscoelasticity: an introduction to mathematical models[END_REF]. In mechanical engineering, there are several applications of fractional calculus, for example, the study of control and dynamical systems [START_REF] Matouk | Dynamical behaviors, linear feedback control and synchronization of the fractional order Liu system[END_REF], modeling anomalous diffusion [START_REF] Wei | A coupled method for inverse source problem of spatial fractional anomalous diffusion equations[END_REF]. The nonstandard behavior or anomaly of the reaction diffusion equation, transport equation, is known as anomalous non Gaussian diffusion, which has long memory effects. Among several explanations for this anomalous diffu- sion, one is by using fractional derivative in time or in space or both in the reaction diffusion equation, transport equation. Anomalous diffusion model solution based on fractional calculus at different operations conditions allow getting better results [START_REF] Metzler | The random walk's guide to anomalous diffusion: a fractional dynamics approach[END_REF]. In physics, there are several potential applications of fractional derivative. A valuable monograph about the applications of fractional integrals and derivatives to polymer physics, biophysics, thermodynamics, rheology, chaotic systems has been edited by R. Hilfer [START_REF] Hilfer | Fractional time evolution[END_REF]. Moreover, in medicine, it has been deduced that the membranes of cells of a biological organism have fractional order electrical conductance and then, they are classified in groups of noninteger order models. Fractional derivatives embody essential features of cell rheological behavior and have enjoyed the greatest success in the field of rheology [START_REF] Arafa | Stability analysis of fractional order hiv infection of CD4+ T cells with numerical solutions[END_REF].

In the literature, several definitions of the fractional derivatives have been proposed. For instance, the Grünwald-Letnikov, the Riemann-Liouville, the Caputo and the Riesz-Feller [START_REF] Podlubny | Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications[END_REF]. Even though they are different, they are all related to each other. The Grünwald-Letnikov is the most obvious approach to define fractional derivatives. It is mainly used for numerical approximation of fractional derivatives. However, dealing with fractional derivatives as a limit of fractional-order difference is not convenient due to the mathematical complexity. Therefore, some studies use the Grünwald Letnikov numerically, but try to solve the initial problem with other definitions. The Riemann-Liouville played an important role for its application in pure mathematics while Caputo has been introduced to respond to applied problems. Indeed, M. Caputo was the first to give applications of fractional calculus to mechanics, especially to linear models of viscoelasticity [START_REF] Caputo | Linear models of dissipation whose Q is almost frequency independent-II[END_REF], [START_REF] Caputo | Elasticitá e dissipazione (Elasticity and anelastic dissipation)[END_REF]. Caputo derivatives allow the use of physically interpretable initial conditions, which is not permitted by the Riemann-Liouville.

Fractional calculus modeling (FCM), using Caputo derivative [START_REF] Sabatier | Advances in fractional calculus[END_REF], has been recently used to generalize the logistic equation. The solution of Chapter 1. Introduction the corresponding fractional differential equation provides a suitable description for the growth of certain types of cancer tumor [START_REF] Varalta | A prelude to the fractional calculus applied to tumor dynamic[END_REF].

A brief historical overview of the development of fractional calculus is given by Ross [START_REF] Ross | A brief history and exposition of the fundamental theory of fractional calculus[END_REF]. The textbook of Oldham and Spanier [START_REF] Oldham | The fractional calculus theory and applications of differentiation and integration to arbitrary order[END_REF] is concerned with the definitions and the properties of fractional order differential/integral operators. In 1987, the huge book was written by Samko, Kilbas, and Marichev [START_REF] Samko | Fractional integrals and derivatives: theory and applications[END_REF], referred to now as "encyclopedia" of fractional calculus. A survey of the many different applications which have emerged from fractional calculus is given in Podlubny [START_REF] Podlubny | Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications[END_REF]. Recently several mathematicians and applied researchers have obtained important results and generalizations from modeling real processes using FC ( [START_REF] Arafa | Stability analysis of fractional order hiv infection of CD4+ T cells with numerical solutions[END_REF], [START_REF] Debnath | Recent applications of fractional calculus to science and engineering[END_REF], [START_REF] Ferreira | Fundamental solution of the multi-dimensional time fractional telegraph equation[END_REF], [START_REF] Ortigueira | What is a fractional derivative[END_REF], [START_REF] Soubhia | Theorem for series in three-parameter Mittag-Leffler function[END_REF]).

Historical review on Inverse Problems

Inverse problems are as old as science itself. A scientific problem is the problem of constructing a model of some physical or biological phenomena that, although inexact, is accurate enough to be able to use observations or measurements to obtain information about the phenomena under investigation. The challenge is to "invert" the model to recover useful estimates of the object under investigation. Strangely enough, given the above description of the scientific method, the mathematical theory of inverse problems was essentially ignored until the middle of the twentieth century. Instead, scientists focused on direct problems, i.e. the construction of the model itself rather than the inversion process. Since the model itself is inexact, such an inversion process typically leads to problems of existence and stability.

By the beginning of the twentieth century, the idea of direct problems dominated mathematical physics. Indeed, the French mathematician Hadamard held the opinion that an important physical problem must be well-posed, i.e. the problem must always have a unique solution that depends continuously on the data. This idea persisted well into the middle of the twentieth century. However, the advent of quantum mechanics and numerous problems in areas of classical physics such as heat conduction and geophysics soon slowly convinced mathematicians and scientists that well-posed direct problems were not the only ones of scientific interest and the mathematical theory of inverse problems began to be developed by mathematicians of the Soviet Union led by Tikhonov. In particular, this theory focused on the problem of determining the parameters and data in the mathematical model of the direct problem from measurements and observations of the data that arise from the physical or biological phenomena taking place.

The solution of an inverse problem is to "invert" the model to recover hidden information about the physical phenomena from the observations.

A comprehensive survey of many areas of inverse problems and imaging can be found in 1600 pages handbook [START_REF] Scherzer | Handbook of mathematical methods in imaging[END_REF].

First publications on inverse and ill-posed problems date back to the first half of the 20th century. Their subjects were related to physics (inverse problems of quantum scattering theory), geophysics (inverse problems of electrical prospecting, seismology, and potential theory), astronomy, and other areas of science. Since the advent of powerful computers, the area of application for the theory of inverse and ill-posed problems has spread to almost all fields of science that uses mathematical methods.

Solving inverse problems can also help to determine the location, shape, and structure of intrusions, defects, sources (of heat, waves, potential difference, pollution), and so on. Given such a wide variety of applications, it is no surprise that the theory of inverse and ill-posed problems have become one of the most rapidly developing areas of modern science since its emergence.

Direct Problems

To define various classes of inverse problems, we should first define a direct (forward) problem. Indeed, something "inverse" must be the opposite of something "direct". Direct problems are based on developing a mathematical model that maps causes into effects and are typically well-posed: each cause has a unique effect and causes which are close to one another have effects which are close to each other. In general, direct problems are well-posed [START_REF] Kirsch | An introduction to the mathematical theory of inverse problems[END_REF]. The well-posedness criteria was proposed by Jacques-Salomon Hadamard, a French mathematician, in 1902. He believed that mathematical models of physical phenomena should have the properties that:

•A solution exists;

•The solution is unique;

•The solution depends continuously on the data (initial conditions and source term).

The first condition describes the consistency of the mathematical model, the second reflects the definiteness of the real situation. The third condition expresses the stability of the equation, a small change in the equation or in the side conditions give rise to a small change in the solution.

In mathematical physics, a direct problem is usually a problem of modeling some physical fields, processes, or phenomena (acoustic, electromagnetic, seismic heat, etc.). The purpose of solving a direct problem is to find a function that describes a physical field or a process at any point of a given domain at any instant of time (if the field is nonstationary). The formulation of a direct problem includes:

•The domain in which the process is studied;

•The equation that describes the process;

•The initial conditions (if the process is nonstationary);

•The conditions on the boundary of the domain (existence and uniqueness involve boundary conditions). The direct problems for fractional diffusion equations such as an initial or boundary value problems have been studied extensively in ( [START_REF] Eidelman | Cauchy problem for fractional diffusion equations[END_REF], [START_REF] Luchko | Some uniqueness and existence results for the initialboundary-value problems for the generalized time-fractional diffusion equation[END_REF], [START_REF] Metzler | Boundary value problems for fractional diffusion equations[END_REF], [START_REF] Sakamoto | Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems[END_REF]) and references therein.

Inverse Problems

An inverse problem is usually ill-posed. The concept of an ill-posed problem is not new. While there is no universal formal definition for inverse problems, Hadamard [1923] defined a problem as being ill-posed if it violates the criteria of a well-posed problem, that is, either existence, uniqueness or continuous dependence on data is no longer true, i.e., arbitrarily small changes in the measurement data lead to indefinitely large changes in the solution. Hadamard did not deal with the numerics of ill-posed problems as he believed that the ill-posedness arose from an incorrect physical representation of the problem.

Most difficulties in solving ill-posed problems are caused by solution instability. Therefore, the term "ill-posed problem" is often used for unstable problems. In the majority of cases, inverse problems turn out to be ill-posed and, conversely, an ill-posed problem can usually be reduced to a problem that is inverse to some direct (well-posed) problem. Inverse and ill-posed problems began to be studied and applied systematically to provide information for many applications in various fields just like physics, geophysics, medicine, astronomy, and all other areas of knowledge where mathematical methods are used. They appear in modeling a wide variety of problems, i.e., Magnetic Resonance Imaging, Computerized Tomography, Signal Processing, and many other applications (see, for example, [START_REF] Bertero | Introduction to inverse problems in imaging[END_REF], [START_REF] Epstein | Introduction to the mathematics of medical imaging[END_REF], [START_REF] Uhlmann | Inside out: inverse problems and applications[END_REF] and [START_REF] Uhlmann | Inverse Problems and Applications: Inside Out II[END_REF]). Solutions of inverse problems recover hidden information for a given system and describe important properties, such as density and velocity of wave propagation, elasticity parameters, conductivity, dielectric permittivity, and magnetic permeability, properties and location of inhomogeneities in inaccessible areas, etc [START_REF] Schumer | Fractional advection-dispersion equations for modeling transport at the Earth surface[END_REF].

One of the first inverse problems solved in the past was Newton's discovery of forces making planets move in accordance with the Kepler's laws. Researches regarding the internal structure of the Earth's crust involved electromagnetic fields in the theory of the inverse problems.

Tichonov [1963] was the first to deal numerically with ill-posedness, and in so doing introduced the concept of regularization. An ill posed problem will often need to be regularized or re-formulated before one can give it a full numerical analysis using computer algorithms or other computational methods. Regularization often involves bringing in new assumptions to fully define the problem and narrow it down. The idea of regularization method is to replace the ill-posed problem by well-posed problem, which can be done by introducing a regularized operator which considers available prior information about the exact solution.

While the concept of regularization is well understood today, the main problem is its implementation in large problems. In recent years signifi- and others. The backward heat equation which is the model of a linear inverse problem is one of the first ill-posed problems that is systematically studied. Solving a heat equation backward in time presents the class of inverse heat conduction problems ( [START_REF] Hon | A fundamental solution method for inverse heat conduction problem[END_REF], [START_REF] Hon | The method of fundamental solution for solving multidimensional inverse heat conduction problems[END_REF], [START_REF] Wei | Reconstruction of a moving boundary from Cauchy data in one-dimensional heat equation[END_REF]). However, there are still a number of unanswered questions, and more importantly, there is insufficient understanding as which method should be used for a specific problem. In the field of nonlinear inverse problems, there are far more advances to be made. Inverse scattering problem for acoustic waves is one of the best-known example of a nonlinear inverse problem and, its electromagnetic version is the mathematical basis of synthetic aperture radar [START_REF] Cheney | Fundamentals of radar imaging[END_REF].

Inverse problems come into various types, for example, inverse initial problems where initial data are unknown and inverse source problems where the source term is unknown. These unknown terms are to be determined using extra boundary data. Fractional differential equations, on the other hand, become an important tool in modeling many real-life problems and hence there has been growing interest in studying inverse problems of time fractional differential equations ( [START_REF] Agarwal | On boundary-value problems for a partial differential equation with Caputo and Bessel operators[END_REF], [START_REF] Aleroev | Determination of a source term for a time fractional diffusion equation with an integral type overdetermining condition[END_REF], [START_REF] Furati | An inverse problem for a generalized fractional diffusion[END_REF], [START_REF] Kirane | An inverse source problem for a two dimensional time fractional diffusion equation with nonlocal boundary conditions[END_REF], [START_REF] Zhang | Inverse source problem for a fractional diffusion equation[END_REF]). The first mathematical results for the inverse problem of finding diffusion coefficient for a fractional differential equation is obtained in [START_REF] Moore | On the uniform convergence of the developments in Bessel functions[END_REF]. Many kinds of boundary problems, including direct and inverse problems, were formulated for the different type of PDEs of integer order and with several fractional order differential operators. For example, in [START_REF] Zhang | Inverse source problem for a fractional diffusion equation[END_REF], Zhang and Xu studied inverse source problem for a fractional diffusion equation where solutions are found based on the method of eigenfunction expansion. Yikan Liu [START_REF] Liu | Strong maximum principle for multi-term time-fractional diffusion equations and its application to an inverse source problem[END_REF] established the strong maximum principle for fractional diffusion equations with multiple Caputo derivatives and investigated the related inverse problem. We also note the work of Daftardar-Gejji and Bhalikar [START_REF] Daftardar-Gejji | Boundary value problems for multiterm fractional differential equations[END_REF] where multi-term fractional diffusion-wave equation was considered and boundary-value problems for this equation were solved by the method of separation of variables. There are many works on studying direct and inverse problems for time-fractional diffusion or diffusion-wave equations with the Caputo derivative. Depending on the operator used in the space-variable, the existence of a classical or generalized solution is partly known, for instance, see works ( [START_REF] Feng | Inverse source problems for time-fractional mixed parabolic-hyperbolic-type equations[END_REF], [START_REF] Luchko | Some uniqueness and existence results for the initialboundary-value problems for the generalized time-fractional diffusion equation[END_REF], [START_REF] Sakamoto | Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems[END_REF]) and references therein. In [START_REF] Masood | Initial inverse problem in heat equation with Bessel operator[END_REF], authors considered the initial inverse problem in heat equation with Bessel operator. They expressed the solution of the problem and the initial temperature distribution in terms of an orthogonal set of Bessel functions. These types of functions arise in the modeling of chemical engineering process including hydrodynamics, bio-processes, diffusion and heat transfer (see, for example, [START_REF] Masood | Initial inverse problem in heat equation with Bessel operator[END_REF], [START_REF] Petrova | Application of Bessel's functions in the modelling of chemical engineering processes[END_REF]). In this thesis, we are interested in the existence Chapter 1. Introduction and uniqueness of solutions of inverse problems for time fractional differential equations.

Presentation of the obtained results

Chapter 2: An inverse problem for a time nonlocal evolution equation with an involution perturbation

Statement of the problem

An inverse problem of a time fractional evolution equation interpolating the heat and wave equations with involution is considered. The goal is to determine the spectral problem associated with our problem and then determine conditions, so that the inverse problem has a unique solution.

The results on the existence and uniqueness of a solution are presented by the method of separation of variables. The equation

D α * u (x, t) -u xx (x, t) + εu xx (-x, t) = f (x) , (1.1)
posed for x ∈ (-π, π) and t > 0, where f and u are unknowns, 1 < α < 2, ε is a nonzero real number such that |ε| < 1. We equip (1.1) with the initial, final, and boundary conditions

u (x, 0) = φ (x) , u t (x, 0) = ρ (x) , u (x, T) = ψ (x) , x ∈ [-π, π] , (1.2) u (-π, t) = 0, u (π, t) = 0, t ∈ [0, T] , (1.3)
where φ (x) and ψ (x) are given sufficiently smooth functions. The derivative D α * defined as

D α * u(x, t) = D α (u(x, t) -u(x, 0) -tu t (x, 0))
is the Caputo derivative for a function built on the Riemann-Liouville derivative D α . Caputo's derivative allows us to impose initial conditions in a natural way. By a regular solution of problem, we mean a pair of functions (u (x, t) , f (x)) of the class u (x, t) ∈ C 2,2 x,t (Ω) ,(space of two times continuously differentiable functions on Ω according to both x and t),

f (x) ∈ C [-π, π],Ω = {-π ≤ x ≤ π, 0 ≤ t ≤ T} .

Main result

The main result of this work is the following theorem.

Theorem 1.3.1 Let φ (x) , ρ (x) , ψ (x) ∈ C 4 [-π, π] and φ (i) (±π) = ρ (i) (±π) = ψ (i) (±π) = 0, i = 0, 1, 2, 3. If 1 -u 0 (.) = 0 then, for a nonzero real number ε such that |ε| < 1, problem (1.1)-(1.2)-(1.
(x, t) = φ (x) + ∞ ∑ k=0 C (4) 1k u 0 (λ 1/α k,2 t) k + 1 2 4 C k+ 1 2 (x) + ∞ ∑ k=0 C (4) 3k u 1 (λ 1/α k,2 t) -C (4) 1k k + 1 2 4 C k+ 1 2 (x) + ∞ ∑ k=1 C (4) 2k u 0 (λ 1/α k,2 t) + C (4) 4k u 1 (λ 1/α k,1 t) -C (4) 2k k 4 S k (x), and
f (x) = ∞ ∑ k=0 (1 -ε) φ (4) 1k -C (4) 1k k + 1 2 2 C k+ 1 2 (x) + ∞ ∑ k=1 (1 + ε) φ (4) 2k -C (4) 2k k 2 S k (x),
where, C

φ (4) 1k ψ (4) 1k

+ C (4) 3k u 1 (λ 1/α k,2 T) 1 -u 0 λ 1/α k,2 T , C (4)
3k = ρ (4) 1k λ 1/α k,2 , C (4) 2k =
φ (4) 2k ψ (4) 2k

+ C (4) 4k u 1 (λ 1/α k,1 T) 1 -u 0 λ 1/α k,1 T , C (4) 4k
= ρ (4) 2k λ 1/α k,1
, and

g (4) 1k = π -π g (4) (x) C k+ 1 2 (x) dx, g (4)
2k = π -π g (4) (x) S k (x) dx, for g = φ, ψ, ρ.

Chapter 3: Some inverse problems for the nonlocal heat equation with Caputo fractional derivative

The purpose of this chapter is to study inverse problems for the nonlocal heat equation with involution of space variable x. Our goal is to determine the spectral problem associated with our problem and then determine conditions for which the inverse problem has a unique solution. The results on the existence and uniqueness of a solution are presented by the method of separation of variables. We consider the heat equation

D α t u (x, t) -u xx (x, t) + εu xx (π -x, t) = f (x) , (1.4) for (x, t) ∈ Ω = {0 < x < π, 0 < t < T < ∞} , 0 < α < 1, where D α t
is the Caputo derivative (which is defined in the next section) and ε is a real number.

Statement of problems

The chapter is concerned with four inverse problems concerning the problem (1.4). We obtain existence and uniqueness results for these problems, based on the Fourier method.

Problem D. Find the couple of functions (u (x, t) , f (x)) satisfying equation (1.4), under the conditions

u (x, 0) = ϕ (x) , x ∈ [0, π] ,
(1.5)

u (x, T) = ψ (x) , x ∈ [0, π] , (1.6)
and the homogeneous Dirichlet boundary conditions

u (0, t) = u (π, t) = 0 , t ∈ [0, T] , (1.7)
where ϕ (x) and ψ (x) are given sufficiently smooth functions.

Problem N. Find the couple of functions (u (x, t) , f (x)) in the domain Ω satisfying equation (1.4), conditions (1.5), (1.6) and the homogeneous Neumann boundary conditions

u x (0, t) = u x (π, t) = 0 , t ∈ [0, T] .
(1.8)

Problem P. Find the couple of functions (u (x, t) , f (x)) in the do- main Ω satisfying equation (1.4), conditions (1.5), (1.6) and the periodic boundary conditions

u (0, t) = u (π, t) , u x (0, t) = u x (π, t) , t ∈ [0, T] .
(1.9)

Problem AP. Find the couple of functions (u (x, t) , f (x)) in the do- main Ω satisfying equation (1.4), conditions (1.5), (1.6) and the antiperiodic boundary conditions

u (0, t) = -u (π, t) , u x (0, t) = -u x (π, t) , t ∈ [0, T] .
(1.10)

A regular solution of problems D, N, P and AP is the pair of functions (u (x, t) , f (x)) where u ∈ C 2,1

x,t ( Ω)(space of two times and one time continuously differentiable functions on Ω according to x and t respectively) and f ∈ C ([0, π]) .

Main results

For the considered problems D, N, P, AP, the following theorems hold true.

Theorem 1.3.2 Let |ε| < 1, ϕ, ψ ∈ C 3 [0, π] and ϕ (i) (0) = ϕ (i) (π) = ψ (i) (0) = ψ (i) (π) = 0, i = 0, 1, 2.
Then the solution of problem D exists, is unique and it can be written in the form u (x, t) = ϕ (x)

+ ∞ ∑ k=0 ( 1-E α,1( -(1-ε)(2k+1) 2 t α )) sin(2k+1)x ( 1-E α,1( -(1-ε)(2k+1) 2 T α )) (2k+1) 2 (ϕ (2) 1k -ψ (2) 1k ) + ∞ ∑ k=1 ( 1-E α,1( -(1+ε)4k 2 t α )) sin 2kx ( 1-E α,1( -(1+ε)4k 2 T α )) 4k 2 (ϕ (2) 2k -ψ (2)
2k ),

f (x) = -ϕ (x) + ε ϕ (π -x) + ∞ ∑ k=0 (1-ε) ϕ (2) 1k -ψ (2) 1k ( 1-E α,1( -(1-ε)(2k+1) 2 T α )) sin(2k + 1)x + ∞ ∑ k=1 (1+ε) ϕ (2) 2k -ψ (2) 2k ( 1-E α,1( -(1+ε)4k 2 T α )) sin 2kx, where ϕ (2) 1k = ϕ (x) , y D 2k+1 , ϕ (2)
2k = ϕ (x) , y D 2k , ψ (2) 1k = ψ (x) , y D 2k+1 , ψ (2)
2k = ψ (x) , y D 2k ,
and E α,β (λt) is the Mittag-Leffler type function:

E α,β (z) = ∞ ∑ m=0 z m Γ(αm + β) . u (x, t) = ϕ (x) + t T (ψ 0 -ϕ 0 ) + ∞ ∑ k=1 ( 1-E α,1 ( -(1-ε)4k 2 t α )) cos 2kx ( 1-E α,1 ( -(1-ε)4k 2 T α )) 4k 2 (ψ (2) 1k -ϕ (2) 1k ) + ∞ ∑ k=0 ( 1-E α,1 ( -(1+ε)(2k+1) 2 t α )) cos(2k+1)x ( 1-E α,1 ( -(1+ε)(2k+1) 2 T α )) (2k+1) 2 (ψ (2) 2k -ϕ (2) 2k ), f (x) = -ϕ (x) + ε ϕ (π -x) + ∞ ∑ k=1 (1-ε) ϕ (2) 1k -ψ (2) 1k ( 1-E α,1 ( -(1-ε)4k 2 T α )) cos 2kx + ∞ ∑ k=0 (1+ε) ϕ (2) 2k -ψ (2) 2k ( 1-E α,1 ( -(1+ε)(2k+1) 2 T α )) cos(2k + 1)x,
where

ϕ 0 = ϕ(x), y N 0 , ϕ (2)
1k = ϕ (x), y N 2k , ϕ (2)
2k = ϕ (x), y N 2k+1 , ψ 0 = ψ(x), y N 0 , ψ (2)
1k = ψ (x), y N 2k , ψ (2)
2k = ψ (x), y N 2k+1 . Theorem 1.3.4 Let ϕ, ψ ∈ C 3 [0, π] and ϕ (i) (0) = ϕ (i) (π) , ψ (i) (0) = ψ (i) (π) , i = 0, 1, 2.
Then the solution of problem P exists, is unique and it can be written in the form

u (x, t) = ϕ (x) + t T (ψ 0 -ϕ 0 ) + ∞ ∑ k=1 ( 1-E α,1 ( -(1-ε)4k 2 t α )) cos 2kx ( 1-E α,1 ( -(1-ε)4k 2 T α )) 4k 2 (ϕ (2) 1k -ψ (2) 1k ) + ∞ ∑ k=1 ( 1-E α,1 ( -(1+ε)4k 2 t α )) sin 2kx ( 1-E α,1 ( -(1+ε)4k 2 T α )) 4k 2 (ϕ (2) 2k -ψ (2) 2k ), 20 Chapter 1. Introduction f (x) = -ϕ (x) + ε ϕ (π -x) + ∞ ∑ k=1 (1-ε) ϕ (2) 1k -ψ (2) 1k ( 1-E α,1 ( -(1-ε)2k 2 T α )) cos 2kx + ∞ ∑ k=1 (1+ε) ϕ (2) 2k -ψ (2) 2k ( 1-E α,1 ( -(1+ε)2k 2 T α )) sin 2kx,
where

ϕ 0 = ϕ (x) , y P 0 , ϕ (2)
1k = ϕ (x) , y P 2k , ϕ (2)
2k = ϕ (x) , y P 2k+1 , ψ 0 = ψ (x) , y P 0 , ψ (2)
1k = ψ (x) , y P 2k , ψ (2)
2k = ψ (x), y P 2k+1 . Theorem 1.3.5 Let ϕ, ψ ∈ C 3 [0, π] and ϕ (i) (0) = -ϕ (i) (π) , ψ (i) (0) = -ψ (i) (π) , i = 0, 1, 2.
Then the solution of problem AP exists, is unique and it can be written in the form

u (x, t) = ϕ (x) + ∞ ∑ k=0 ( 1-E α,1 ( -(1-ε)(2k+1) 2 t α )) cos(2k+1)x ( 1-E α,1 ( -(1-ε)(2k+1) 2 T α )) (2k+1) 2 (ϕ (2) 1k -ψ (2) 1k ) + ∞ ∑ k=0 ( 1-E α,1 ( -(1+ε)(2k+1) 2 t α )) sin(2k+1)x ( 1-E α,1 ( -(1+ε)(2k+1) 2 T α )) (2k+1) 2 (ϕ (2) 2k -ψ (2) 2k ), f (x) = -ϕ (x) + ε ϕ (π -x) + ∞ ∑ k=0 (1-ε) ϕ (2) 1k -ψ (2) 1k ( 1-E α,1 ( -(1-ε)(2k+1) 2 T α )) cos(2k + 1)x + ∞ ∑ k=0 (1+ε) ϕ (2) 2k -ψ (2) 2k ( 1-E α,1 ( -(1+ε)(2k+1) 2 T α )) sin(2k + 1)x,
where

ϕ (2) 1k = ϕ (x) , y AP 2k+1 , ϕ (2)
2k = ϕ (x) , y AP 2k , ψ (2) 1k = ψ (x), y AP 2k+1 , ψ (2)
2k = ψ (x) , y AP 2k .

Chapter 4: Inverse source problems for a wave equation with involution

The purpose of this chapter is to study inverse problems for a nonlocal wave equation with involution of space variable x. We consider the nonlocal wave equation

u tt (x, t) -u xx (x, t) + εu xx (π -x, t) = f (x) , (1.11)
for (x, t) ∈ Ω = {0 < x < π, 0 < t < T} , where ε is a real number.

Statement of problems

The chapter is devoted to two inverse problems concerning the wave equation with a perturbative term of involution type with respect to the space variable. We obtain existence and uniqueness results for these problems, based on the Fourier method.

Problem D. Find a couple of functions (u (x, t) , f (x)) satisfying equa- tion (1.11), under the conditions

u (x, 0) = 0, x ∈ [0, π] ,
(1.12)

u (x, T) = ψ (x) , x ∈ [0, π] , (1.13)
u t (x, 0) = 0, x ∈ [0, π] , (1.14)
and the homogeneous Dirichlet boundary conditions

u (0, t) = u (π, t) = 0 , t ∈ [0, T] , (1.15)
where ψ(x) is a given sufficiently smooth function.

Problem N. Find the couple of functions (u (x, t) , f (x)) in the do- main Ω satisfying equation (1.11), conditions (1.12), (1.13), (1.14) and the homogeneous Neumann boundary conditions

u x (0, t) = u x (π, t) = 0 , t ∈ [0, T] .
(1.16)

A regular solution of the problems D and N is the pair of functions (u (x, t) , f (x)) , where u ∈ C 2 ( Ω) and f ∈ C ([0, π]) .

Spectral properties of the perturbed Sturm-Liouville problem

Application of the Fourier method for solving problems D and N leads to a spectral problem defined by the equation

y (x) -εy (π -x) + λy (x) = 0, 0 < x < π, (1.17)
and one of the following boundary conditions

y (0) = y (π) = 0, (1.18)
y (0) = y (π) = 0. (1.19)

Main results

For the considered problems D and N, the following theorems are valid.

Theorem 1 Let |ε| < 1, ψ ∈ C 4 [0, π] and ψ (i) (0) = ψ (i) (π) = 0, i = 0, 1, 2, 3, 4. If cos √ 1 -ε (2k + 1) T < δ 1 < 1 and cos √ 1 + ε2kT < δ 2 < 1,
then the solution of problem D exists, is unique and it can be written in the form 21) where ψ

u (x, t) = ∞ ∑ k=0 1 -cos √ 1 -ε (2k + 1) t sin (2k + 1) x 1 -cos √ 1 -ε (2k + 1) T (2k + 1) 4 ψ 4 2k+1 + ∞ ∑ k=1 1 -cos √ 1 + ε2kt sin 2kx 1 -cos √ 1 + ε2kT 16k 4 ψ 4 2k , (1.20) f (x) = ∞ ∑ k=0 (1 -ε) ψ 4 2k+1 1 -cos √ 1 -ε (2k + 1) T (2k + 1) 2 sin (2k + 1) x + ∞ ∑ k=1 (1 + ε) ψ 4 2k 1 -cos √ 1 + ε2kT 4k 2 sin 2kx, (1.
(4) 2k+1 = ψ (4) (x) , y D 2k+1 and ψ (4) 2k = ψ (4) (x) , y D 2k . Theorem 2 Let |ε| < 1, ψ ∈ C 4 [0, π] and ψ (i) (0) = ψ (i) (π) = 0, i = 0, 1, 2, 3, 4. If cos √ 1 -ε (2k + 1) T < σ 1 < 1 and cos √ 1 + ε2kT < σ 2 < 1,
then the solution of problem N exists, is unique and it can be written in the form 23) where ψ

u (x, t) = ∞ ∑ k=0 1 -cos √ 1 + ε (2k + 1) t cos (2k + 1) x 1 -cos √ 1 + ε (2k + 1) T (2k + 1) 4 ψ 4 2k+1 + ∞ ∑ k=1 1 -cos √ 1 -ε2kt cos 2kx 1 -cos √ 1 -ε2kT 16k 4 ψ 4 2k , (1.22) f (x) = ∞ ∑ k=0 (1 + ε) ψ 4 2k+1 1 -cos √ 1 + ε (2k + 1) T (2k + 1) 2 cos (2k + 1) x + ∞ ∑ k=1 (1 -ε) ψ 4 2k 1 -cos √ 1 -ε2kT 4k 2 cos 2kx, (1.
(4) 2k+1 = ψ (4) (x) , y N 2k+1 and ψ (4) 2k = ψ (4) (x) , y N 2k .

Chapter 5: On a nonlinear boundary-layer problem for the fractional Blasius type equation

Statement of the problem and main results

In this chapter, we consider a non-linear sequential differential equation with Caputo fractional derivative. We reduce the problem to the equivalent nonlinear integral equation and we prove the complete continuity of the nonlinear integral operator. Consider the boundary value problem for the nonlinear fractional differential equation of Blasius type

d dt D 2α * x (t) + M (x (t) , t) D 2α * x (t) = 0, a < t < b, (1.24)
with boundary conditions

x (a) = ϕ 1 , D α * x (a) = ϕ 2 , x (b) = ϕ 3 , (1.25)
where α ∈ 1 2 , 1 and ϕ 1 , ϕ 2 , ϕ 3 are given real numbers. Note that when α = 1, problem (1.24) -(1.25) is met in boundary layer theory in fluid mechanics and polymer theory. The recent surge in developing the theory of fractional differential equations has motivated the present work.

Condition (*).

Let M (x, t) be defined and continuous in the domain

G = {(x, t) : |x| ≤ R, R > 0, a ≤ t ≤ b} , where R = |ϕ 2 | (b -a) α Γ (α + 1) + |ϕ 1 | + ϕ 3 - ϕ 2 (b -a) α Γ (α + 1) -ϕ 1 ,
and m = min x,t∈G M (x, t) , M = max max x,t∈G M (x, t) , 0 .
The space C 3 3-α ([a, b]) denotes the space:

C 3 3-α ([a, b]) = x ∈ C ([a, b]) : x ∈ C 3-α ([a, b]) .
Here

C 3-α ([a, b]) = (t -a) 3-α x ∈ C ([a, b]) .
The main result of this work is the following theorem. ) . Theorem on the existence of a unique solution of the problem for the non-linear differential equation of fractional order is formulated. In the limiting case, the considered boundary problem coincides with the boundary-layer problem for the Blasius equation.

Preliminaries

Basic Functions

In this section, we recall definitions of some special functions that we use later in the thesis.

Gamma Function

The Euler's gamma function Γ (z) is one of the basic functions of fractional calculus. It generalizes the factorial z! to take also non-integers and complex values and it is defined as follows.

Definition 1.4. [START_REF] Agarwal | On boundary-value problems for a partial differential equation with Caputo and Bessel operators[END_REF] The gamma function Γ (.) is defined as: for z ∈ C and Re (z) > 0

Γ (z) = ∞ 0 e -t t z-1 dt (2.1)
where t z-1 = e (z-1) log(t) . This integral is convergent for all complex z ∈ C (Re (z) > 0) .

For this function the reduction formula

Γ (z + 1) = zΓ (z) (Re (z) > 0)
holds. In particular, if z = n ∈ N 0 , then

Γ (n + 1) = n! (n ∈ N 0 )
with (as usual) 0! = 1.

Mittag-Leffler functions

Definition 1.4. [START_REF] Ahmad | On nonlocal boundary value problems for nonlinear integrodifferential equations of arbitrary fractional order[END_REF] The classical Mittag-Leffler function is defined by Mittag-Leffler (1903) :

E α (z) = ∞ ∑ n=0 z n Γ (nα + 1)
, z ∈ C and Re (α) > 0.

(2.2)

Taking α = 1 the exponential function is recovered, E 1 (z) = e z . A two parameter generalization has been proposed by Wiman (1905) as follows:

E α,β (z) = ∞ ∑ n=0 z n Γ (αn + β)
, z ∈ C and Re (α) , Re (β) > 0.

(2.3)

For β = 1 the classical Mittag-Leffler function is recovered, i.e. E α,1 (z) = E α (z) .

Fractional Derivatives

Often the easiest access to the idea of the non-integer differential and integral operators studied in the field of fractional calculus is given by Cauchy's well known representation of an n-fold integral as a convolution integral:

I n a f (t) = t a s n-1 a ... s 1 a f (s) dsds 1 ds n-1 = 1 (n -1)! t a (t -s) n-1 f (s) ds, n ∈ N, t ∈ R + , (2.4)
where I n a is the n-fold integral operator (Cauchy formula). The Riemann-Liouville fractional integral is a simple generalization of the Cauchy formula (2.4) , the integer n is substituted by a positive real number α and the Gamma function Γ (.) is used instead of the factorial, i.e.

I α a f (t) = 1 Γ (α) t α (t -s) α-1 f (s) ds, α, t ∈ R + .
(

The definition of fractional integral is very straightforward and there are no complications. A more difficult question is how to define a fractional derivative.

We can give the simplest definition of fractional derivative as concatenation of integer order differentiation and fractional integration, i.e.

D α a f (t) = D n I n-α a f (t) or C D α a f (t) = I n-α a D n f (t) ,
where n is the integer satisfying α ≤ n < α + 1 and D n , n ∈ N, is the n-fold differential operator. The operator D α a is usually denoted as Riemann-Liouville differential operator, while the operator C D α a is named Caputo differential operator.

Riemann-Liouville definition

The Riemann-Liouville fractional derivative of a function f ∈

AC n ([a, b]), where -∞ < a < b < +∞, with α, t ∈ R + is defined as follow: a D α t f (t) = 1 Γ (n -α) d n dt n t a f (s) ds (t -s) α-n+1 , (n -1 < α ≤ n) , n ∈ N. (2.6)
The Riemann-Liouville derivative has certain disadvantages when trying to model real world phenomena with fractional differential equations. Therefore, we shall introduce a modified fractional differential operator proposed by Caputo ( [21], [START_REF] Diethelm | The analysis of fractional differential equations: An application-oriented exposition using differential operators of Caputo type[END_REF]).

M. Caputo definition

The Caputo fractional derivative of a function f ∈

C n ([a, b]), where -∞ < a < b < +∞, with α, t ∈ R + is defined by, C a D α t f (t) = 1 Γ (n -α) t a f (n) (s) ds (t -s) α-n+1 , (n -1 < α ≤ n) , n ∈ N. (2.7)
The Caputo derivative allows the use of physically interpretable initial conditions, which is not permitted by the Riemann-Liouville. Another difference between the two definitions appears when dealing with constant function. Indeed, for a constant, the Caputo fractional derivative is zero while Reamann-Liouville fractional derivative is not zero.

Relation between Riemann-Liouville and Caputo derivatives

Proposition 1.4.3 If f (t) is n -1 continuously differentiable in the interval [a, b] and f (n) (t) is integrable in [a, b] , then R D α t f (t) = m-1 ∑ i=0 f (i) (a) (t -a) i-α Γ (1 + i -α) + C D α t f (t) (2.8)
where m -1 ≤ α ≤ m < n with m ∈ N * .

Proof. Applying repeatedly integration by parts to the Riemann-Liouville will give us: for ∀t ∈ R, and α < t,

R D α t f (t) = m-1 ∑ i=0 f (i) (a) (t -a) i-α Γ (1 + i -α) + 1 Γ (m -α) t a f (m) τdτ (t -τ) α-m+1 .
(2.9)

The right hand side of the above equation is equal to the Caputo derivatives.

Proposition 1.4. [START_REF] Akhmanov | Generation of structures in optical systems with two-dimensional feedback: On the way to the creation of nonlinear optical analogues of neural networks[END_REF] Let f (t) be n -

1 continuously differentiable in the in- terval [a, b] and f (n) (t) be integrable in [a, b] . Then, if f (n) (a) = 0, for n = 0, 1, 2, ..., m -1, R D α t f (t) = C D α t f (t) for any t ∈ R, (2.10)
where m -1 ≤ α ≤ m < n with m ∈ N * .

Properties of Fractional Derivatives

We recall some useful properties of fractional derivatives.

1. Linearity

Preliminaries 27

Assuming that the fractional derivatives of f and g exists, then for λ, µ ∈ R [START_REF] Podlubny | Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications[END_REF] :

D α (λ f (t) + µg (t)) = λD α f (t) + µD α g (t) , (2.11)
where D α denotes any of the fractional derivatives we have defined before.

Laplace Transform of Fractional Derivatives

We assume that the fractional derivative and the Laplace transform of f exists. Then, the Laplace transform of the Riemann-Liouville and Caputo fractional derivative are defined by [START_REF] Podlubny | Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications[END_REF] :

∀s ∈ C, -Riemann-Liouville L [ R D α t f (t)] = s α f (s) - n-1 ∑ i=0 s i R D α-i-1 t f (t) t=0 , n -1 ≤ α < n (2.12) -Caputo L [ C D α t f (t)] = s α f (s) - n-1 ∑ i=0 s α-i-1 f (i) (0) , n -1 ≤ α < n (2.13)
where f (.) denotes the Laplace transform of f (.) , and s is the variable in the frequency domain.

Fourier Transform of Fractional Derivatives

We assume that the fractional derivative and the Fourier transform of f exists. Then, the Fourier transform of the Grünwald-Letnikov, Riemann-Liouville, and the Caputo fractional derivatives are defined by [START_REF] Agarwal | On boundary-value problems for a partial differential equation with Caputo and Bessel operators[END_REF]

: ∀k ∈ R, F [ O D α t f (t)] = (-ik) α f (s) (2.14)
where O D α t denotes any of the mentioned fractional differentiations, f (.) denotes the Fourier transform of f (.) , and k is the variable in the fre- quency domain.

Derivative of the Fractional Operator with Respect to α.

In the next proposition, we present the derivative of the fractional derivative with respect to the fractional order α. We consider the left Riemann-Liouville derivative. However, similar results can be obtained using other definitions.

Proposition 1.4.5 [START_REF] Podlubny | Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications[END_REF] If the α th order Riemann-Liouville derivative of f exists where n -1 ≤ α < n, then the derivative of ∂ α f ∂x α with respect to α is given by

∂ ∂α ∂ α f (x) ∂x α = ψ 0 (n -α) ∂ α f (x) ∂x α - 1 Γ (n -α) d n dx n x 0 (x -τ) n-α-1 ln (x -τ) f (τ) dτ, (2.15) where ψ 0 (n -α) = Γ (n-α) Γ(n-α) .
Chapter 1. Introduction

Proof. The result can be obtained by differentiating (2.6) with respect to α.

30

Chapter 2. An inverse problem for time fractional evolution equations with an involution perturbation

Introduction and statement of the problem

Differential equations with operations (equations with shift (involution)), apparently started with the work of Babbage [START_REF] Babbage | An essay towards the calculus of functions. Part II[END_REF], and were discussed later by Carleman [START_REF] Carleman | Sur la théorie des équations intégrales et ses applications[END_REF] in 1932. In the late sixties and early seventies of the 20th century, Przewoerska-Rolewicz addressed many questions about differential equations with involutions in a series of nice papers [START_REF] Przeworska-Rolewicz | On equations with several involutions of different orders and its applications to partial differential-difference equations[END_REF], [START_REF] Przeworska-Rolewicz | Sur les équations involutives et leurs applications[END_REF], [START_REF] Przeworska-Rolewicz | On equations with reflection[END_REF], [START_REF] Przeworska-Rolewicz | On equations with rotations[END_REF], [START_REF] Przeworska-Rolewicz | On linear differential equations with transformed argument solvable by means of right invertible operators[END_REF], [START_REF] Przeworska-Rolewicz | Right invertible operators and functionaldifferential equations with involutions[END_REF] and then compiled her results in form of a text [START_REF] Przeworska-Rolewicz | Equations with transformed argument: an algebraic approach[END_REF]. For generalized solutions of functional differential equations, see the book by Wiener [START_REF] Wiener | Generalized solutions of functional differential equations[END_REF]. Recently, Kaliev et al [START_REF] Kaliev | Inverse problem for forward-backward parabolic equation with generalized conjugation conditions[END_REF], [START_REF] Kaliev | Problems of determining the temperature and density of heat sources from the initial and final temperatures[END_REF], Orazov and Sadybekov [START_REF] Orazov | One nonlocal problem of determination of the temperature and density of heat sources[END_REF], [START_REF] Orazov | On a class of problems of determining the temperature and density of heat sources given initial and final temperature[END_REF],Sarsenbi [START_REF] Sarsenbi | Unconditional bases related to a nonclassical secondorder differential operator[END_REF], Sadybekov and Sarsenbi [START_REF] Sadybekov | On the notion of regularity of boundary value problems for differential equation of second order with dump argument[END_REF], Sarsenbi and Tengaeva [START_REF] Sarsenbi | On the basis properties of root functions of two generalized eigenvalue problems[END_REF], treated spectral problems and inverse problems for evolution equations with involution. In their talk [START_REF] Cabada | Equations with involutions[END_REF], Cabada and Tojo mentioned an application of a parabolic equation with an involution related to heat conduction. In this chapter, we address an inverse problem for the time fractional evolution equation with involution

D α * u (x, t) -u xx (x, t) + εu xx (-x, t) = f (x) , (2.1)
posed for x ∈ (-π, π) and t > 0, where f and u are unknowns, 1 < α < 2, ε is a nonzero real number such that |ε| < 1. We equip (2.1) with the initial, final, and boundary conditions

u (x, 0) = φ (x) , u t (x, 0) = ρ (x) , u (x, T) = ψ (x) , x ∈ [-π, π] , (2.2) u (-π, t) = 0, u (π, t) = 0, t ∈ [0, T] , (2.3)
where φ (x) and ψ (x) are given sufficiently smooth functions. The derivative D α * defined as

D α * u(x, t) = D α (u(x, t) -u(x, 0) -tu t (x, 0))
is the Caputo derivative for a regular function built on the Riemann-Liouville derivative D α . Caputo's derivative allows us to impose initial conditions in a natural way. By a regular solution of the problem, we mean a pair of functions (u (x, t) , f (x)) of the class u (x, t) ∈ C 2,2

x,t (Ω)(space of two times con- tinuously differentiable functions on Ω according to both x and t),

f (x) ∈ C [-π, π], Ω = {-π ≤ x ≤ π, 0 ≤ t ≤ T} .
When one uses the method of separation of variables to solve the problem, a spectral problem appears, which is mentioned in the next section.

The Spectral Problem

The spectral problem consists of the equation:

X (x) -εX (-x) + λX (x) = 0, -π ≤ x ≤ π, (2.4)
where λ is the spectral parameter, equipped with the boundary conditions:

X (-π) = X (π) = 0, X (-π) = X (π) = 0. (2.5)
It is proved in [START_REF] Kopzhassarova | Basis properties of eigenfunctions of second-order differential operators with involution[END_REF] that expressing the solution of problem (2.4)-(2.5) in terms of the sum of even and odd functions, one finds the following eigenvalues:

λ k,1 = (1 + ε) k 2 , k ∈ Z + , λ k,2 = (1 -ε) k + 1 2 2 , k ∈ N,
with the corresponding normalized eigenfunctions given by

X k,1 = 1 √ π sin kx =: S k (x), k ∈ Z + , X k,2 = 1 √ π cos k + 1 2 x =: C k+ 1 2 (x), k ∈ N. (2.6)
Observe that the systems of functions (2.6) is complete in L 2 (-π, π) . [59] We will use the following result which appears in part 3 of [START_REF] Gorenflo | Fractional calculus. In Fractals and fractional calculus in continuum mechanics[END_REF].

Lemma 2.2. [START_REF] Agarwal | On boundary-value problems for a partial differential equation with Caputo and Bessel operators[END_REF] The following differential equation of fractional order α > 0

D α * u(t) = D α u(t) - m-1 ∑ k=0 t k k! u (k) (0 + ) = -u(t) + q(t), t > 0,
where m is a positive integer uniquely defined by m -1 < α ≤ m, with the prescribed initial values

u (k) (0 + ) = c k , k = 0, 1, 2, ..., m -1, has the solution u(t) = m-1 ∑ k=0 c k u k (t) + t 0 q(t -s)u δ (s) ds, with u k (t) = J k u 0 (t), u (h) k = δ k,h , h, k = 0, 1, ..., m -1, (2.7) u δ (t) = -u 0 (t),
the functions u k (t) represent the fundamental solutions of the differential equation of order m.

Main results

Here we present the existence and uniqueness results for our problem.

Chapter 2. An inverse problem for time fractional evolution equations with an involution perturbation

Theorem 2.3.1 Let φ (x) , ρ (x) , ψ (x) ∈ C 4 [-π, π] and φ (i) (±π) = ρ (i) (±π) = ψ (i) (±π) = 0, i = 0, 1, 2, 3. If 1 -u 0 (.) = 0 then, for a nonzero real number ε such that |ε| < 1, problem (2.1) -(2.
3) has a unique solution which can be written in the form u (x, t) = φ (x)

+ ∞ ∑ k=0 C (4) 1k u 0 (λ 1/α k,2 t) k + 1 2 4 C k+ 1 2 (x) + ∞ ∑ k=0 C (4) 3k u 1 (λ 1/α k,2 t) -C (4) 1k k + 1 2 4 C k+ 1 2 (x) + ∞ ∑ k=1 C (4) 2k u 0 (λ 1/α k,2 t) + C (4) 4k u 1 (λ 1/α k,1 t) -C (4) 2k k 4 S k (x), and f (x) = ∞ ∑ k=0 (1 -ε) φ (4) 1k -C (4) 1k k + 1 2 2 C k+ 1 2 (x) + ∞ ∑ k=1 (1 + ε) φ (4) 2k -C (4) 2k k 2 S k (x),
where C (4) 1k = φ (4) 1k ψ (4) 1k

+ C (4) 3k u 1 (λ 1/α k,2 T) 1 -u 0 λ 1/α k,2 T , C (4
) 3k = ρ (4) 1k λ 1/α k,2 , C (4
) 2k = φ (4) 2k -ψ (4) 2k + C (4) 4k u 1 (λ 1/α k,1 T) 1 -u 0 λ 1/α k,1 T , C (4
) 4k = ρ (4) 2k λ 1/α k,1
, and g (4) 1k

= π -π g (4) (x) C k+ 1 2 (x) dx, g (4)

Existence

Here, we give the full proof of the existence of a solution of the problem as stated in Theorem 2.3.1. As the eigenfunctions system (2.6) forms an orthonormal basis in L 2 (-π, π), the functions u (x, t) and f (x) can be represented as follows

u(x, t) = ∞ ∑ k=0 u k (t)C k+ 1 2 (x) + ∞ ∑ k=1 v k (t) S k (x),
(2.9)

and

f (x) = ∞ ∑ k=0 f 1k C k+ 1 2 (x) + ∞ ∑ k=1 f 2k S k (x), (2.10)
where u k (t) , v k (t) , f 1k and f 2k are unknown. Substituting (2.9) and (2.10) into equation (2.1), we obtain the following equations for the functions u k (t) , v k (t) and the constants f 1k , f 2k :

D α * u k (t) + λ k,2 u k (t) = f 1k , and
D α * v k (t) + λ k,1 v k (t) = f 2k , which we write 1 λ k,2 D α * u k (t) + u k (t) = 1 λ k,2 f 1k , (2.11)
and 1

λ k,1 D α * v k (t) + v k (t) = 1 λ k,1 f 2k .
(2.12)

By a change of scale t → λ 1 α k,2 t in (2.11), and using Lemma 2.2.1, we obtain

u k (t) = c 0 u 0 (λ 1 α k,2 t) + c 1 u 1 (λ 1 α k,2 t) - f 1k λ k,2
(u 0 (λ

1 α k,2 t) -u 0 (0 + )),
where

u 0 (τ) = e α (τ) = E α (-τ α ) = ∞ ∑ n=0 (-τ α ) n Γ(αn + 1)
,

u 1 (τ) = Je α (τ) = t 0 e α (s) ds, u 0 (0 + ) = 1, u 0 (0 + ) = 0, u 1 (0 + ) = 0, u 1 (0 + ) = 1,
(e α and Je α represent the fundamental solution of the equation

D α * u k (t) = -u k (t) + f 1k /λ k,2 );
Chapter 2. An inverse problem for time fractional evolution equations with an involution perturbation so

u k (t) = (c 0 - f 1k λ k,2 )u 0 (λ 1 α k,2 t) + c 1 u 1 (λ 1 α k,2 t) + f 1k λ k,2 ,
and

u k (t) =: C 1k u 0 (λ 1 α k,2 t) + C 3k u 1 (λ 1 α k,2 t) + f 1k λ k,2 .
Similarly

v k (t) = (c 0 - f 2k λ k,1
)u 0 (λ

1 α k,1 t) + c 1 u 1 (λ 1 α k,1 t) + f 1k λ k,1 ,
and

v k (t) =: C 2k u 0 (λ 1 α k,1 t) + C 4k u 1 (λ 1 α k,1 t) + f 2k λ k,1 ,
where the constants C 1k , C 2k , C 3k , C 4k , f 1k , and f 2k are to be determined using the given data.

Expanding the functions φ (x) , ρ (x) and ψ (x) using the eigenfunc- tions system (2.6), we obtain

C 1k = φ 1k - f 1k λ k,2 , C 2k = φ 2k - f 2k λ k,1 , λ 1/α k,2 C 3k = ρ 1k , λ 1/α k,1 kC 4k = ρ 2k , C 1k u 0 (λ 1/α k,2 T) + C 3k u 1 (λ 1/α k,2 T) + f 1k λ k,2 = ψ 1k ,
and

C 2k u 0 (λ 1/α k,1 T) + C 4k u 1 (λ 1/α k,1 T) + f 2k λ k,1 = ψ 2k ,
where, φ ik , ρ ik , ψ ik , i = 1, 2 are the coefficients of the expansions of the functions φ (x) , ρ (x) , ψ (x) given by

g 1k = π -π g (x) C k+ 1 2 (x) dx, g 2k = π -π g (x) S k (x) dx, for g = φ, ρ, ψ.
Solving the above set of equations for C 1k , C 2k , C 3k , C 4k , f 1k , and f 2k , we get

C 1k = φ 1k -ψ 1k + C 3k u 1 (λ 1/α k,2 T) 1 -u 0 (λ 1/α k,2 T) , C 3k = ρ 1k λ 1/α k,2
,

C 2k = φ 2k -ψ 2k + C 4k u 1 (λ 1/α k,1 T) 1 -u 0 (λ 1/α k,1 T) , C 4k = ρ 2k λ 1/α k,1
, and

f 1k = λ k,2 (φ 1k -C 1k ) , f 2k = λ k,1 (φ 2k -C 2k ) . (2.13)
Now, substituting the expressions for u k (t) , v k (t) , f 1k , f 2k into (2.9) and (2.10), we obtain

u (x, t) = φ(x) + ∞ ∑ k=0 C 1k u 0 (λ 1/α k,2 t) + C 3k u 1 (λ 1/α k,2 t) -C 1k C k+ 1 2 (x) + ∞ ∑ k=1 C 2k u 0 (λ 1/α k,1 t) + C 4k u 1 (λ 1/α k,1 t) -C 2k S k x, and
f (x) = ∞ ∑ k=0 λ k,2 (φ 1k -C 1k ) C k+ 1 2 (x) + ∞ ∑ k=1 λ k,1 (φ 2k -C 2k ) S k (x). Moreover, if φ (i) (±π) = ρ (i) (±π) = ψ (i) (±π) = 0, i = 0, 1, 2, 3, then integrating φ ik , ρ ik , ψ ik , i = 1, 2
, by parts yields

g 1k = g (4) 1k k + 1 2 4
and g 2k = g (4) 2k k 4 , for g = φ, ρ, ψ, where, φ (4) ik , ρ (4 ik , ψ (4) ik , i = 1, 2 are the coefficients of the expansions of the functions φ (4) (x) , ρ (4) (x) , ψ (4) (x) and are given by (g (4) )

1k = π -π g (4) (x)C k+ 1 2 (x) dx, (g (4) ) 2k = π -π g (4) (x)S k (x) dx, for g = φ, ψ, ρ.
Then the constants C 1k , C 2k , C 3k , C 4k , f 1k , and f 2k can be written as

C 1k = C (4) 1k k + 1 2 4 , C 3k = C (4) 3k k + 1 2 4 , C 2k = C (4) 2k k 4 , C 4k = C (4) 4k k 4 ,
and

f 1k = (1 -ε) k + 1 2 2 φ (4) 1k -C (4) 1k , f 2k = (1 + ε) k 2 φ (4) 2k -C (4) 2k , (2.14)
Chapter 2. An inverse problem for time fractional evolution equations with an involution perturbation where,

C (4) 1k = φ (4) 1k -ψ (4) 1k + C (4) 3k u 1 (λ 1/α k,2 T) 1 -u 0 (λ 1/α k,2 T) , C (4) 3k = (ρ (4) ) 1k λ 1/α k,2 , C (4)
2k = (φ (4) ) 2kψ (4) 2k

+ C (4) 4k u 1 (λ 1/α k,1 T) 1 -u 0 (λ 1/α k,1 T) , C (4) 4k = (ρ (4) ) 2k λ 1/α k,1
.

Thus the solution of our problem takes the form

u (x, t) = φ (x) + ∞ ∑ k=0 C (4) 1k u 1 (λ 1/α k,2 t) (k + 1 2 ) 4 C k+ 1 2 (x) + ∞ ∑ k=0 C (4) 3k u 1 (λ 1/α k,2 t) -C (4) 1k k + 1 2 4 C k+ 1 2 (x) + ∞ ∑ k=1 C (4) 2k u 0 (λ 1/α k,1 t) + C (4) 4k u 1 (λ 1/α k,1 t) -C (4) 2k k 4 S k (x),
(2.15) and

f (x) = ∞ ∑ k=0 (1 -ε) φ (4) 1k -C (4) 1k k + 1 2 2 C k+ 1 2 (x) + ∞ ∑ k=1 (1 + ε) φ (4) 2k -C (4) 2k k 2 S k (x). (2.16)
This completes the proof of Theorem 2.3.1.

Convergence of the series

To establish that the formal solution is indeed a true solution, we will show that all operations performed in the proof are valid.

The convergence of the series in (2.15) and (2.16) are based on the following estimates for u (x, t) and f (x):

|u(x, t)| ≤ |φ(x)| + 2 √ π ∞ ∑ k=0 √ 2 φ (4) 1k + √ 2 ψ (4) 1k + 4 ρ (4) 1k √ 1 -ε 1 -u 0 (λ 1/α k,2 T) k + 1 2 4 + 2 √ π ∞ ∑ k=1 √ 2 φ (4) 2k + √ 2 ψ (4) 2k + 2 ρ (4) 2k √ 1 + ε 1 -u 0 λ 1/α k,1 T k 4
(2.17) and

| f (x)| ≤ 1 √ π ∞ ∑ k=0 6 φ (4) 1k + 2 ψ (4) 1k + 2 √ 2 ρ (4) 1k 1 -u 0 (λ 1/α k,2 T) k + 1 2 2 + 1 √ π ∞ ∑ k=1 6 φ (4) 2k + 2 ψ (4) 2k + √ 2 ρ (4) 2k 1 -u 0 (λ 1/α k,1 T) k 2 (2.18) As φ (x) , ρ (x) , ψ (x) ∈ C 4 [-π, π],
by the Bessel inequality for trigono- metric series, the following series converge:

∞ ∑ k=0 g (4) 1k 2 ≤C g (4) (x) 2 L 2 (-π,π)
, for g = φ, ρ, ψ.

(2. [START_REF] Cabada | Positive solutions of nonlinear fractional differential equations with integral boundary value conditions[END_REF] and

∞ ∑ k=1 g (4) 2k 2 ≤C g (4) (x) 2 L 2 (-π,π)
, for g = φ, ρ, ψ, (2.20) which implies that the set

ϕ (4) ik , ρ (4)
ik , ψ

(4) ik , k = 1, 2.
is bounded.

Theorem 2.4.1 (Weierstrass M test) Let {u n } be a sequence of real or complex- valued functions defined on a set X and that there is a sequence of positive numbers {M n } satisfying ∀n ≥ 1, ∀x ∈ X:

| f n (x)| ≤ M n , such that ∞ ∑ n=1 M n < ∞ Then the series ∞ ∑ k=1
f n (x) converges absolutely and uniformly on X.

Therefore, by the Weierstrass M-test, series (2.17) and (2.18) converge absolutely and uniformly in the region Ω. Now, using termwise differentiation of the series (2.15) twice with respect to the variables x and t, we get the following estimates for u xx (x, t) and u tt (x, t),

|u xx (x, t)| ≤ φ (x) + 2 √ π ∞ ∑ k=0 √ 2 φ (4) 1k + √ 2 ψ (4) 1k + 4 ρ (4) 1k √ 1 -ε 1 -u 0 (λ 1/α k,2 T) k + 1 2 2 + 2 √ π ∞ ∑ k=1 √ 2 φ (4) 2k + √ 2 ψ (4) 2k + 2 ρ (4) 2k √ 1 + ε 1 -u 0 (λ 1/α k,1 T) k 2
Chapter 2. An inverse problem for time fractional evolution equations with an involution perturbation and (2.20) and the Weierstrass M-test, also converge absolutely and uniformly on Ω.

|u tt (x, t)| ≤ 1 √ π ∞ ∑ k=0 2 φ (4) 1k + 2 ψ (4) 1k + 6 √ 2 ρ (4) 1k 1 -u 0 (λ 1/α k,2 T) k + 1 2 2 + 1 √ π ∞ ∑ k=1 2 φ (4) 2k + 2 ψ (4) 2k + 3 √ 2 ρ (4) 2k 1 -u 0 (λ 1/α k,1 T) k 2 which, on using (2.19),

Uniqueness

The uniqueness of the solution follows from representation of the solution given in the theorem, and from the completeness of the system (2.6).

Suppose that there are two solutions {u 1 (x, t) ,

f 1 (x)} and {u 2 (x, t) , f 2 (x)} of problem. Denote u (x, t) = u 1 (x, t) -u 2 (x, t) and f (x) = f 1 (x) -f 2 (x) .
Then the functions u (x, t) and f (x) satisfy (2.1), conditions of (2.2) and homogeneous boundary conditions(2.3).

Let

u k (t) = π -π u (x, t) S k (x)dx, k ∈ Z + , (2.21)
v k (t) = π -π u (x, t) C k+ 1 2 (x)dx, k ∈ N, (2.22)
f 1k = π -π f (x) S k (x)dx, k ∈ Z + , (2.23)
f 2k = π -π f (x) C k+ 1 2 (x)dx, k ∈ N. (2.24)
Applying the operator D α to the equation (2.21) we have

D α u k (t) = π -π D α t u (x, t) S k (x)dx = π -π (u xx (x, t) -εu xx (-x, t)) S k (x)dx + f 1k .
Integrating by parts and taking into account the homogeneous conditions (2.2) and (2.3) , we obtain

D α u k,1 (t) = f 1k , u (0) = 0 , u (T) = 0. Consequently, f 1k = 0, u k (t) ≡ 0.
In a similar way for the functions (2.22), (2.23), (2.24) one proves that

f 2k = 0, v k (t) ≡ 0.
Further, by the completeness of the system (2.6) in L 2 (-π, π) we ob- tain f (t) ≡ 0, u (x, t) ≡ 0, 0

≤ t ≤ T, -π ≤ x ≤ π.
Uniqueness of the solution of problem is proved.

Introduction

The purpose of this chapter is to study inverse problems for the nonlocal heat equation with involution of space variable x. We consider the heat equation

D α t u (x, t) -u xx (x, t) + εu xx (π -x, t) = f (x) , (3.1)
for (x, t) ∈ Ω = {0 < x < π, 0 < t < T < ∞} , 0 < α < 1, where D α t is the Caputo derivative (which is defined in the next section) and ε is a nonzero real number such that |ε| < 1. Before describing our results, let us dwell a while on the existing literature concerning differential equation with delay either in time or in space.

Differential equations with time deviating arguments have been treated in a sizable number of articles and monographs, to cite but a few: [START_REF] Ismagilov | Examples of very unstable linear partial functional differential equations[END_REF], [START_REF] Kuang | Delay differential equations: with applications in population dynamics[END_REF], [START_REF] Wu | Theory and applications of partial functional differential equations[END_REF]. For example in [START_REF] Ismagilov | Examples of very unstable linear partial functional differential equations[END_REF], the authors considered an example of parabolic functional differential equation with time delay of the following form:

u t = u xx (t -h, x) , t > 0, 0 < x < π, h > 0,
to study the spectrum distribution of its symbols (characteristic quasipolinomials). Delay differential equations occur in a variety of real world applications: biological modelling, automatic control systems, economics, epidemiology, feedback problems, the theory of climate models, etc.

Ample opportunities of applying equations with deviating argument in mathematical models have increased the interest of the study of new problems for partial differential equations [START_REF] Bzheumikhova | Application of Fourier method to investigation of the Dirichlet problem for partial differential equations with deviating arguments[END_REF], [START_REF] Hernández | A note on partial functional differential equations with state-dependent delay[END_REF], [START_REF] Rezounenko | Stability of positive solutions of local partial differential equations with a nonlinear integral delay term1[END_REF]. Among differential equations with deviating arguments, special place is occupied by equations with a deviation of arguments of alternating character. Such deviations include the so-called deviation of involution type [START_REF] Cabada | General Results for Differential Equations with Involutions[END_REF]. To describe them, let Γ be an interval in R and let X ∈ Γ be a real variable. The homeomorphism

α 2 (X) = α (α (X)) = X,
is called a Carleman shift (deviation of involution) [START_REF] Carleman | Sur la théorie des équations intégrales et ses applications[END_REF].

Equations containing Carleman shift are equations with an alternating deviation (at X * < X being equations with advanced, and at X * > X being equations with delay, where X * is a fixed point of the mapping α (X) ).

However, some interesting works contain equations with modifications of the spatial variable in the unknown function that are motivated by the nonlinear optics, studied in a number of papers (see [START_REF] Klevchuk | Bifurcation of the state of equilibrium in the system of nonlinear parabolic equations with transformed argument[END_REF], [START_REF] Muravnik | On the asymptotics of the solution of the Cauchy problem for some differential-difference parabolic equations[END_REF]); for instance, in [START_REF] Muravnik | On the asymptotics of the solution of the Cauchy problem for some differential-difference parabolic equations[END_REF], the author studied the Cauchy problem for the difference-differential parabolic equation where M is a finite set of vectors in R n parallel to coordinate axes ( or any other orthogonal vector system) and the coefficients a h are real. It should be noted that parabolic functional-differential equation arise in the investigation of nonlinear optic systems with two-dimensional feedback(see [START_REF] Akhmanov | Generation of structures in optical systems with two-dimensional feedback: On the way to the creation of nonlinear optical analogues of neural networks[END_REF] and the references therein). Skubachevskii introduced these operators for nonlocal problems in heat conduction problems [START_REF] Skubachevskii | First mixed problem for a parabolic difference-differential equation[END_REF]. In contrast to classical parabolic differential equations, these equations have a number of new properties. For instance, the smoothness of generalized solution can be violated inside the cylindrical domain even for an infinitely smooth right-hand side of the equation.

∂u ∂t = ∆u + ∑ h∈M a h u(x -h, t),
Furthermore, for the equations containing transformation of the spatial variable in the diffusion term , we can cite the talk of Cabada and Tojo [START_REF] Tojo | Equations with involutions[END_REF], where an example that describes a concrete situation in physics is given: Consider a metal wire around a thin sheet of insulating material in a way that some parts overlap some others as shown in Figure 1.

Assuming that the position y = 0 is the lowest of the wire, and the insulation goes up to the left at -Y and to the right up to Y.

For the proximity of two sections of wires they added the third term with modifications on the spatial variable to the right-hand side of the heat equation with respect to the wire:

∂T ∂t (y, t) = α ∂ 2 T ∂y 2 (y, t) + β ∂ 2 T ∂y 2 (-y, t),
where T is the temperature at (y, t). Such equations have also a purely theoretical value.

Concerning the inverse problems and spectral problems for equations with involutions, some recent works have been done by Kaliev [START_REF] Kaliev | Inverse problem for forward-backward parabolic equation with generalized conjugation conditions[END_REF], [START_REF] Sabitova | Problems of determining the temperature and density of heat sources from the initial and nal temperatures[END_REF], Kirane [START_REF] Kirane | Inverse problems for a nonlocal wave equation with an involution perturbation[END_REF], [START_REF] Al-Salti | On a class of inverse problems for a heat equation with involution perturbation[END_REF], Sadybekov [START_REF] Orazov | One nonlocal problem of determination of the temperature and density of heat sources[END_REF], [START_REF] Orazov | On a class of problems of determining the temperature and density of heat sources given initial and final temperature[END_REF], Sarsenbi [START_REF] Kopzhassarova | Basis properties of eigenfunctions of second-order differential operators with involution[END_REF], [START_REF] Sarsenbi | Unconditional bases related to a nonclassical secondorder differential operator[END_REF], [START_REF] Sarsenbi | On the basis properties of root functions of two generalized eigenvalue problems[END_REF].

Problem N. Find the couple of functions (u (x, t) , f (x)) in the domain Ω satisfying equation (3.1), conditions (3.2), (3.3) and the homogeneous Neumann boundary conditions

u x (0, t) = u x (π, t) = 0 , t ∈ [0, T] . (3.5)
Problem P. Find the couple of functions (u (x, t) , f (x)) in the do- main Ω satisfying equation (3.1), conditions (3.2), (3.3) and the periodic boundary conditions

u (0, t) = u (π, t) , u x (0, t) = u x (π, t) , t ∈ [0, T] . (3.6)
Problem AP. Find the couple of functions (u (x, t) , f (x)) in the do- main Ω satisfying equation (3.1), conditions (3.2), (3.3) and the antiperiodic boundary conditions

u (0, t) = -u (π, t) , u x (0, t) = -u x (π, t) , t ∈ [0, T] . (3.7)
A regular solution of problems D, N, P and AP is the pair of functions (u (x, t) , f (x)) where u ∈ C 2,1

x,t ( Ω)(space of two times and one time continuously differentiable functions on Ω according to x and t respectively)and f ∈ C ([0, π]) .

Note that similar problems for the heat equation and their fractional analogues have been considered in [START_REF] Furati | An inverse problem for a generalized fractional diffusion[END_REF], [START_REF] Kirane | Determination of an unknown source term and the temperature distribution for the linear heat equation involving fractional derivative in time[END_REF], [START_REF] Nguyen | Regularized solution of an inverse source problem for a time fractional diffusion equation[END_REF].

Spectral properties of the perturbed Sturm-Liouville problem

Application of the Fourier method for solving problems D, N, P, AP leads to the spectral problem defined by the equation

y (x) -εy (π -x) + λy (x) = 0, 0 < x < π, ( 3.8)
and one of the following boundary conditions

y (0) = y (π) = 0, (3.9)
y (0) = y (π) = 0, (3.10)
y (0) = y (π) , y (0) = y (π) , (3.11)
y (0) = -y (π) , y (0) = -y (π) . (3.12)
It is easy to see that the Sturm-Liouville problem for the equation (3.8) with one of the boundary conditions (3.9) -(3.12) is self-adjoint. It is known that the self-adjoint problem has real eigenvalues and their eigenfunctions form a complete orthonormal basis in L 2 (0, π) [75]. To further investigate the problems under consideration, we need to calculate the explicit form of the eigenvalues and eigenfunctions.

For |ε| < 1 the problem (3.8), (3.9) has the following eigenvalues

λ D 2k = (1 + ε) 4k 2 , k ∈ N, λ D 2k+1 = (1 -ε) (2k + 1) 2 , k ∈ N 0 = N ∪ {0} and eigenfunctions        y D 2k = 2 π sin(2kx), k ∈ N, y D 2k+1 = 2 π sin (2k + 1) x, k ∈ N 0 . (3.13)
Similarly, the problem (3.8), (3.10) has the eigenvalues

λ N 2k+1 = (1 + ε) (2k + 1) 2 , k ∈ N 0 , λ N 2k = (1 -ε) 4k 2 , k ∈ N 0 , and corresponding eigenfunctions                  y N 0 = 1 √ π , y N 2k+1 = 2 π cos (2k + 1) x, k ∈ N 0 , y N 2k = 2 π cos(2kx), k ∈ N. (3.14)
The eigenvalues of the problem (3.8), (3.11) are

λ P 2k+1 = (1 + ε) 4k 2 , k ∈ N, λ P 2k = (1 -ε) 4k 2 , k ∈ N 0 ,
Chapter 3. Some inverse problems for the nonlocal heat equation with Caputo fractional derivative with the corresponding eigenfunctions

                 y P 0 = 1 √ π , y P 2k+1 = 2 π sin(2kx), k ∈ N, y P 2k = 2 π cos(2kx), k ∈ N. (3.15)
Finally, the problem (3.8), (3.12) has the following eigenvalues Proof It is known (see. [START_REF] Moiseev | The basis property for systems of sines and cosines[END_REF]) that the systems of (3.15) and (3.16) form a complete orthonormal system in L 2 (0, π). It remains to prove the completeness of systems (3.13) and (3.14). We prove the completeness of the system (3.13).

λ AP 2k+1 = (1 + ε) (2k + 1) 2 , k ∈ N 0 , λ AP 2k = (1 -ε) (2k + 1) 2 , k ∈ N 0 , and corresponding eigenfunctions        y AP 2k+1 = 2 π cos (2k + 1) x, k ∈ N 0 , y AP 2k = 2 π sin (2k + 1) x, k ∈ N 0 . ( 3
The system (3.13) is complete in L 2 (0, π) if the equalities

π 0 f (x) sin(2kx)dx = 0, k ∈ N, π 0 f (x) sin (2k + 1) xdx = 0, k ∈ N 0 , for f ∈ L 2 (0, π) lead to f (x) = 0 in L 2 (0, π).
We have

π 0 f (x) sin (2k + 1) xdx = π 2 0 f (x) sin (2k + 1) xdx + π π 2 f (x) sin (2k + 1) xdx = π 2 0 ( f (x) -f (π -x)) sin (2k + 1) xdx = 0. ( 3
.17) Then by the completeness of the system {sin (2k + 1) x} k∈N 0 in L 2 0, π 2 [START_REF] Moiseev | The basis property for systems of sines and cosines[END_REF] , we obtain

f (x) = f (π -x), 0 < x < π 2 . Similarly π 0 f (x) sin (2kx)dx = π 2 0 f (x) sin(2kx)dx + π π 2 f (x) sin(2kx)dx = π 2 0 ( f (x) + f (π -x)) sin(2kx)dx = 0.

Main results

47

Then by the completeness of the system {sin(2kx)} k∈N in L 2 0, π 2 [START_REF] Moiseev | The basis property for systems of sines and cosines[END_REF], we have f (x) =f (πx), 0 < x < π 2 . Whereupon , f (x) = 0 in L 2 0, π 2 , and consequently f (x) = 0 in L 2 (0, π). The completeness of the system (3.14) is proved similarly.

Main results

For the considered problems D, N, P, AP, the following theorems hold true.

Theorem 3.4.1 Let |ε| < 1, ϕ, ψ ∈ C 3 [0, π] and ϕ (i) (0) = ϕ (i) (π) = ψ (i) (0) = ψ (i) (π) = 0, i = 0, 1, 2.
Then the solution of the problem D exists, is unique and it can be written in the form

u (x, t) = ϕ (x) + ∞ ∑ k=0 ( 1-E α,1 ( -(1-ε)(2k+1) 2 t α )) sin(2k+1)x ( 1-E α,1 ( -(1-ε)(2k+1) 2 T α )) (2k+1) 2 (ϕ (2) 1k -ψ (2) 1k ) + ∞ ∑ k=1 ( 1-E α,1 ( -(1+ε)4k 2 t α )) sin(2kx) ( 1-E α,1 ( -(1+ε)4k 2 T α )) 4k 2 (ϕ (2) 2k -ψ (2) 2k ), f (x) = -ϕ (x) + ε ϕ (π -x) + ∞ ∑ k=0 (1-ε) ϕ (2) 1k -ψ (2) 1k ( 1-E α,1 ( -(1-ε)(2k+1) 2 T α )) sin(2k + 1)x + ∞ ∑ k=1 (1+ε) ϕ (2) 2k -ψ (2) 2k ( 1-E α,1 ( -(1+ε)4k 2 T α )) sin(2kx),
where

ϕ (2) 1k = ϕ (x) , y D 2k+1 , ϕ (2) 2k = ϕ (x) , y D 2k , ψ (2) 1k = ψ (x) , y D 2k+1 , ψ (2) 2k = ψ (x) , y D 2k ,
and E α,β (λt) is the Mittag-Leffler type function:

E α,β (z) = ∞ ∑ m=0 z m Γ(αm + β) . f (x) = -ϕ (x) + ε ϕ (π -x) + ∞ ∑ k=1 (1-ε) ϕ (2) 1k -ψ (2) 1k ( 1-E α,1 ( -(1-ε)2k 2 T α )) cos(2kx) + ∞ ∑ k=1 (1+ε) ϕ (2) 2k -ψ (2) 2k ( 1-E α,1 ( -(1+ε)2k 2 T α )) sin(2kx),
where

ϕ 0 = ϕ (x) , y P 0 , ϕ (2)
1k = ϕ (x) , y P 2k , ϕ (2)
2k = ϕ (x) , y P 2k+1 , ψ 0 = ψ (x) , y P 0 , ψ (2)
1k = ψ (x) , y P 2k , ψ (2)
2k = ψ (x), y P 2k+1 .

Theorem 3.4.4 Let ϕ, ψ ∈ C 3 [0, π] and ϕ (i) (0) = -ϕ (i) (π) , ψ (i) (0) = -ψ (i) (π) , i = 0, 1, 2.
Then the solution of problem AP exists, is unique and it can be written in the form

u (x, t) = ϕ (x) + ∞ ∑ k=0 ( 1-E α,1 ( -(1-ε)(2k+1) 2 t α )) cos(2k+1)x ( 1-E α,1 ( -(1-ε)(2k+1) 2 T α )) (2k+1) 2 (ϕ (2) 1k -ψ (2) 1k ) + ∞ ∑ k=0 ( 1-E α,1 ( -(1+ε)(2k+1) 2 t α )) sin(2k+1)x ( 1-E α,1 ( -(1+ε)(2k+1) 2 T α )) (2k+1) 2 (ϕ (2) 2k -ψ (2) 2k ), f (x) = -ϕ (x) + ε ϕ (π -x) + ∞ ∑ k=0 (1-ε) ϕ (2) 1k -ψ (2) 1k ( 1-E α,1 ( -(1-ε)(2k+1) 2 T α )) cos(2k + 1)x + ∞ ∑ k=0 (1+ε) ϕ (2) 2k -ψ (2) 2k ( 1-E α,1 ( -(1+ε)(2k+1) 2 T α )) sin(2k + 1)x,
where

ϕ (2) 1k = ϕ (x) , y AP 2k+1 , ϕ (2)
2k = ϕ (x) , y AP 2k , ψ (2) 1k = ψ (x), y AP 2k+1 , ψ (2)
2k = ψ (x) , y AP 2k .

50

Chapter 3. Some inverse problems for the nonlocal heat equation with Caputo fractional derivative

Proof of the existence of the solution of problem D

We give the full proof for problem D. The existence of the solution of problems P, N and AP are proved analogously.

As the eigenfunctions system (3.13) of problem D forms an orthonormal basis in L 2 (0, π) (this follows from the self-adjoint problem (3.8), (3.9)), the functions u (x, t) and f (x) can be expanded as follows

u (x, t) = ∞ ∑ k=0 u k (t) sin (2k + 1) x + ∞ ∑ k=1 v k (t) sin(2kx), (3.18) f (x) = ∞ ∑ k=0 f 1 k sin (2k + 1) x + ∞ ∑ k=1 f 2 k sin(2kx), (3.19)
where f 1 k , f 2 k , u k (t) , v k (t) are unknown. Substituting (3.18) and (3.19) into (3.1), we obtain the following equation for the functions u k (t) , v k (t) and the constants

f 1 k , f 2 k : D α u k (t) + (1 -ε) (2k + 1) 2 u k (t) = f 1 k , D α v k (t) + (1 + ε) 4k 2 v k (t) = f 2 k
. Solving these equations [START_REF] Kilbas | Theory and applications of fractional differential equations[END_REF] we obtain

u k (t) = f 1 k (1 -ε) (2k + 1) 2 + C 1k E α,1 -(1 -ε) (2k + 1) 2 t α , v k (t) = f 2 k (1 + ε) k 2 + C 2k E α,1 -(1 + ε) 4k 2 t α ,
where the constants C 1k , C 2k , f 1 k , f 2 k are unknown. To find these constants, we use conditions (3.2). Let ϕ ik , ψ ik , i = 1, 2 be the coefficients of the expansions of ϕ (x) and ψ (x) We first find C 1k :

ϕ 1k = 2 π π 0 ϕ (x) sin (2k + 1) xdx, ϕ 2k = 2 π π 0 ϕ (x) sin(2kx)dx, ψ 1k = 2 π π 0 ψ (x) sin (2k + 1) xdx, ψ 2k = 2 π π 0 ψ (x) sin(2kx)dx.
u k (0) = f 1 k (1 -ε) (2k + 1) 2 + C 1k = ϕ 1k , u k (T) = f 1 k (1 -ε) (2k + 1) 2 + C 1k E α,1 -(1 -ε) (2k + 1) 2 T α = ψ 1k , ϕ 1k -C 1k + C 1k E α,1 -(1 -ε) (2k + 1) 2 T α = ψ 1k .
Then

C 1k = ϕ 1k -ψ 1k 1 -E α,1 -(1 -ε) (2k + 1) 2 T α . The constant f 1
k is represented as

f 1 k = (1 -ε) (2k + 1) 2 ϕ 1k -(1 -ε) (2k + 1) 2 C 1k .
Now we find C 2k :

v k (0) = f 2 k (1 + ε) 4k 2 + C 2k = ϕ 2k , v k (T) = f 2 k (1 + ε) 4k 2 + C 2k e -(1+ε)4k 2 T = ψ 2k , ϕ 2k -C 2k + C 2k e -(1+ε)4k 2 T = ψ 2k . Then C 2k = ϕ 2k -ψ 2k 1 -E α,1 (-(1 + ε) 4k 2 T α ) .
For the constant f 2 k , we found :

f 2 k = (1 + ε) 4k 2 ϕ 2k -(1 + ε) 4k 2 C 2k . Substituting u k (t) , v k (t) , f 1 k , f 2 k into (3.18) and (3.19) we find u (x, t) = ϕ (x) + ∞ ∑ k=0 C 1k E α,1 -(1 -ε) (2k + 1) 2 t α -1 sin(2k + 1)x + ∞ ∑ k=1 C 2k E α,1 -(1 + ε) 4k 2 t α -1 sin(2kx).
Suppose that

ϕ (i) (0) = 0 , ϕ (i) (π) = 0 , i = 0, 1, 2,
ψ (i) (0) = 0 , ψ (i) (π) = 0 , i = 0, 1, 2, Caputo fractional derivative and f (x) = f 1 (x) -f 2 (x) .
Then the functions u (x, t) and f (x) satisfy (3.1)-(3.3) and periodic con- ditions(3.6).

Let

u 0 (t) = 1 √ π π 0 u (x, t) dx, (3.24)
u 1k (t) = 2 π π 0 u (x, t) cos(2kx)dx, k ∈ N, (3.25)
u 2k (t) = 2 π π 0 u (x, t) sin(2kx)dx, k ∈ N, (3.26)
f 0 = 1 √ π π 0 f (x)dx, (3.27)
f 1k = 2 π π 0 f (x) cos(2kx)dx, k ∈ N, (3.28)
f 2k = 2 π π 0 f (x) sin(2kx)dx, k ∈ N. (3.29)
Applying the operator D α to the equation (3.24) we have

D α u 0 (t) = 1 √ π π 0 D α t u (x, t) dx = 1 √ π π 0 (u xx (x, t) -εu xx (π -x, t)) dx + f 0 .
Integrating by parts and taking into account the homogeneous conditions (3.2) and (3.6) , we obtain

D α u 0 (t) = f 0 , u (0) = 0 , u (T) = 0.
Consequently, f 0 = 0, u 0 (t) ≡ 0. In a similar way for the functions (3.25), (3.26), (3.27), (3.28), (3.29) one proves that f 1k = 0, f 2k = 0, u 1k (t) ≡ 0, u 2k (t) ≡ 0.

Further, by the completeness of the system (3.15) in L 2 (0, π) we obtain f (t) ≡ 0, u (x, t) ≡ 0, 0 ≤ t ≤ T, 0 ≤ x ≤ π.

Uniqueness of the solution of problem P is proved. The uniqueness of the solution of problems D, N and AP can be proved similarly.

Introduction

In many physical problems, determination of coefficients or righthand side according to some available information (the source term, in case of a wave equation) in a differential equation is required; these problems are known as inverse problems. These kinds of problems are illposed in the sense of Hadamard.

The purpose of this chapter is to study inverse problems for a nonlocal wave equation with involution of space variable x. We consider the nonlocal wave equation

u tt (x, t) -u xx (x, t) + εu xx (π -x, t) = f (x) , (4.1)
for (x, t) ∈ Ω = {0 < x < π, 0 < t < T} , where ε is a real number.

Wide opportunities for applying equations with deviating argument in mathematical models have increased the interest of the study of new problems for partial differential equations [START_REF] Bzheumikhova | Application of Fourier method to investigation of the Dirichlet problem for partial differential equations with deviating arguments[END_REF], [START_REF] Hernández | A note on partial functional differential equations with state-dependent delay[END_REF], [START_REF] Rezounenko | Stability of positive solutions of local partial differential equations with a nonlinear integral delay term1[END_REF].

Equations with a deviation of arguments of alternating character has special interest in differential equations with deviating arguments. Such deviations include the so-called deviation of involution type [START_REF] Cabada | Differential equations with involutions[END_REF]. To describe them, let Γ be an interval in R and let X ∈ Γ be a real variable.

The homeomorphism

α 2 (X) = α (α (X)) = X
is called a Carleman shift (deviation of involution) [START_REF] Carleman | Sur la théorie des équations intégrales et ses applications[END_REF].

Equations containing Carleman shift are equations with an alternating deviation (at X * < X being equations with advanced, and at X * > X being equations with delay, where X * is a fixed point of the mapping α (X) ).

Concerning the inverse problems for partial differential equations with involutions, some recent works have been implemented in [START_REF] Ahmad | An inverse problem for space and time fractional evolution equations with an involution perturbation[END_REF][START_REF] Al-Salti | On a class of inverse problems for a heat equation with involution perturbation[END_REF][START_REF] Kirane | Inverse problems for a nonlocal wave equation with an involution perturbation[END_REF][START_REF] Kirane | Determination of an unknown source term temperature distribution for the sub-diffusion equation at the initial and final data[END_REF][START_REF] Torebek | Some inverse problems for the nonlocal heat equation with Caputo fractional derivative[END_REF].

Statement of problems

The chapter is devoted to two inverse problems concerning the wave equation with a perturbative term of involution type with respect to the space variable. We obtain existence and uniqueness results for these problems, based on the Fourier method.

Problem D. Find a couple of functions (u (x, t) , f (x)) satisfying equa- tion (4.1), under the conditions

u (x, 0) = 0, x ∈ [0, π] , (4.2) u (x, T) = ψ (x) , x ∈ [0, π] , (4.3) u t (x, 0) = 0, x ∈ [0, π] , (4.4
(0, t) = u (π, t) = 0 , t ∈ [0, T] , (4.5)
where ψ(x) is a given sufficiently smooth function.

Problem N. Find the couple of functions (u (x, t) , f (x)) in the domain Ω satisfying equation (4.1), conditions (4.2), (4.3), (4.4) and the homogeneous Neumann boundary conditions

u x (0, t) = u x (π, t) = 0 , t ∈ [0, T] . ( 4.6)
A regular solution of the problems D and N is the pair of functions (u (x, t) , f (x)) , where u ∈ C 2 ( Ω) and f ∈ C ([0, π]) .

Spectral properties of the perturbed Sturm-Liouville problem

Application of the Fourier method for solving problems D and N leads to a spectral problem defined by the equation

y (x) -εy (π -x) + λy (x) = 0, 0 < x < π, ( 4.7)
and one of the following boundary conditions

y (0) = y (π) = 0, (4.8)
y (0) = y (π) = 0. (4.9)

It is easy to see that the Sturm-Liouville problem for the equation (4.7) with one of the boundary conditions (4.8) and (4.9) is self-adjoint. It is known that the self-adjoint problem has real eigenvalues and their eigenfunctions form a complete orthonormal basis in L 2 (0, π) [75]. To further investigate the problems under consideration, we need to calculate the explicit form of the eigenvalues and eigenfunctions.

It is easy to show that for |ε| < 1 the problem (4.7), (4.8) has the following eigenvalues (4.15) where ψ

λ D 2k = (1 + ε) 4k 2 , k ∈ N, λ D 2k+1 = (1 -ε) (2k + 1) 2 , k ∈ N 0 = N ∪ {0} and eigenfunctions        y D 2k = 2 π sin 2kx, k ∈ N, y D 2k+1 = 2 π sin (2k + 1) x, k ∈ N 0 . (4.10) u (x, t) = ∞ ∑ k=0 1 -cos √ 1 + ε (2k + 1) t cos (2k + 1) x 1 -cos √ 1 + ε (2k + 1) T (2k + 1) 4 ψ 4 2k+1 + ∞ ∑ k=1 1 -cos √ 1 -ε2kt cos 2kx 1 -cos √ 1 -ε2kT 16k 4 ψ 4 2k , (4.14) f (x) = ∞ ∑ k=0 (1 + ε) ψ 4 2k+1 1 -cos √ 1 + ε (2k + 1) T (2k + 1) 2 cos (2k + 1) x + ∞ ∑ k=1 (1 -ε) ψ 4 2k 1 -cos √ 1 -ε2kT 4k 2 cos 2kx,
(4) 2k+1 = ψ (4) (x) , y N 2k+1 and ψ (4) 2k = ψ (4) (x) , y N 2k .

Proof of the uniqueness of the solution

Suppose that there are two solutions {u

1 (x, t) , f 1 (x)} and {u 2 (x, t) , f 2 (x)} of the problem N. Denote u (x, t) = u 1 (x, t) -u 2 (x, t) and f (x) = f 1 (x) -f 2 (x) .
Then the functions u (x, t) and f (x) satisfy (4.1)-(4.4)and homogenious conditions (4.6).

Let

u 0 (t) = 1 √ π π 0 u (x, t) dx, (4.16
)

u 2k (t) = 2 π π 0 u (x, t) cos 2kxdx, k ∈ N, (4.17)
u 2k+1 (t) = 2 π π 0 u (x, t) cos(2k + 1)xdx, k ∈ N 0 , (4.18)
f 0 = 1 √ π π 0 f (x)dx, (4.19
)

f 2k = 2 π π 0 f (x) cos 2kxdx, k ∈ N, (4.20)
f 2k+1 = 2 π π 0 f (x) cos(2k + 1)xdx, k ∈ N. (4.21)
Applying the operator ∂ 2 ∂t 2 to the equation (4.16) we have

u 0 (t) = 1 √ π π 0 u tt (x, t) dx = 1 √ π π 0 (u xx (x, t) -εu xx (π -x, t)) dx + f 0 .
Integrating by parts and taking into account the homogeneous conditions (4.2) and (4.6) , we obtain

u 0 (t) = f 0 , u (0) = 0 , u (T) = 0, u (0) = 0.
Hence it is easy to get f 0 = 0, u 0 (t) ≡ 0. In a similar way for the functions (4.17), (4.18), (4.19), (4.20), (4.21) it is easy to prove that

f 2k = 0, f 2k+1 = 0, u 2k (t) ≡ 0, u 2k+1 (t) ≡ 0.
Further, by the completeness of the system (4.10) in L 2 (0, π) we obtain

f (x) ≡ 0, u (x, t) ≡ 0, 0 ≤ t ≤ T, 0 ≤ x ≤ π.
The uniqueness of the solution of the problem N is proved. The uniqueness of the solution of the problem D can be proved similarly.

Proof of the existence of the solution

We give the full proof for the problem D. The existence of the solution of the problem N is proved analogously. As the eigenfunctions system (4.10) of the problem D forms an orthonormal basis in L 2 (0, π) (this follows from the self-adjoint problem (4.7), (4.8)), the functions u (x, t) and f (x) can be expanded as follows

u (x, t) = ∞ ∑ k=0 u 2k+1 (t) sin (2k + 1) x + ∞ ∑ k=1 u 2k (t) sin 2kx, (4.22) f (x) = ∞ ∑ k=0 f 2k+1 sin (2k + 1) x + ∞ ∑ k=1 f 2k sin 2kx, (4.23)
where f 2k+1 , f 2k , u 2k+1 (t) , u 2k (t) are unknown. Substituting (4.22) and (4.23) into (4.1), we obtain the following equation for the functions u 2k+1 (t) , u 2k (t) and the constants f 2k+1 , f 2k :

u 2k+1 (t) + (1 -ε) (2k + 1) 2 u 2k+1 (t) = f 2k+1 , u 2k (t) + (1 + ε) 4k 2 u 2k (t) = f 2k .
Solving these equations [START_REF] Kilbas | Theory and applications of fractional differential equations[END_REF], we obtain

u 2k+1 (t) = f 2k+1 (1 -ε) (2k + 1) 2 + C 1k cos √ 1 -ε (2k + 1) t + C 2k sin √ 1 -ε (2k + 1) t, u 2k (t) = f 2k (1 + ε) 4k 2 + D 1k cos √ 1 + ε2kt + D 2k sin √ 1 + ε2kt,
where the constants C 1k , C 2k , D 1k , D 2k , f 2k+1 , f 2k are unknown. To find these constants, we use the conditions (4.2). Let ψ 2k , ψ 2k+1 be the coefficients of the expansions of ψ (x)

ψ 2k+1 = 2 π π 0 ψ (x) sin (2k + 1) xdx, ψ 2k = 2 π π 0 ψ (x) sin 2kxdx.
We first find C 1k , C 2k :

u 2k+1 (0) = f 2k+1 (1 -ε) (2k + 1) 2 + C 1k = 0, u 2k+1 (0) = C 2k = 0, u 2k+1 (T) = f 2k+1 (1 -ε) (2k + 1) 2 1 -cos √ 1 -ε (2k + 1) T = ψ 2k+1 .
The constant f 2k+1 is represented as

f 2k+1 = (1 -ε) (2k + 1) 2 ψ 2k+1 1 -cos √ 1 -ε (2k + 1) T .
Now we find D 1k , D 2k :

u 2k (0) = f 2k (1 + ε) 4k 2 + D 1k = 0, u 2k (0) = D 2k = 0, u 2k (T) = f 2k (1 + ε) 4k 2 1 -cos √ 1 + ε2kT = ψ 2k .
For the constant f 2k , we find:

f 2k = (1 + ε) 4k 2 ψ 2k 1 -cos √ 1 + ε2kT .
On a nonlinear boundary-layer problem for the fractional blasius type equation

Abstract

In this chapter, we consider a non-linear sequential differential equation with Caputo fractional derivative of Blasius type and we reduce the problem to the equivalent nonlinear integral equation. We prove the complete continuity of the nonlinear integral operator. The theorem on the existence of a solution of the problem for the Blasius equation of fractional order is also proved.

Introduction

Various fields of science and engineering deal with dynamical systems, which can be described by fractional-order equations. Recently, many authors have studied fractional-order differential equations from two aspects: the theoretical aspects of existence and uniqueness of solutions and the analytic and numerical methods for finding solutions. The interest in the study of fractional order differential equations lies in the fact that fractional-order models in some situations are found to be more accurate than the classical integer-order models, that is, there are more degrees of freedom in the fractional-order models.

It is well known that in fluid mechanics, the problems are mostly described by systems of partial differential equations (PDEs). If somehow, a system can be reduced to a single ordinary differential equation (ODE), this constitutes a considerable mathematical simplification of the problem. For this goal, one of the approaches is based on the introduction of new variables having the form of dimensionless combinations of the initially given physical variables. Therefore, if the number of independent variables can be reduced, then PDEs can be replaced by ODEs. In the problem of the modelling of boundary layer, this is sometimes possible, and in some cases, the system of PDEs reduces to a system involving a third order differential equation of the form

f + f f + g f = 0, (5.1)
where g : (-∞, ∞) -→ (-∞, ∞) is some function. Notice that equation (5.1) with g (x) = β x 2 -1 was first introduced by Falkner and Skan in their classical work [START_REF] Falkneb | Solutions of the boundary-layer equations[END_REF] and for this reason is called the Falkner-Skan equation.

The most famous example for these types of equations is the Blasius equation:

f + f f = 0, (5.2)
which corresponds to g(x) = 0 and arises in the study of the laminar boundary layer on a flat plate. For more information see Brighi [START_REF] Brighi | The Equation f"'+ ff"+ g (f')= 0 and the Associated Boundary Value Problems[END_REF] and the references therein. It is well known [START_REF] Blasius | The boundary layers in fluids with little friction[END_REF] that the Blasius equation (5.2) with conditions:

f (0) = 0, f (0) = 0, f (∞) = 1
has a unique solution. Note also that in [START_REF] Falkneb | Solutions of the boundary-layer equations[END_REF] the author proves some important results concerning to the so-called subsolutions and supersolutions of the Blasius equation (5.2).

In this chapter, we consider a non-linear sequential differential equation with Caputo fractional derivative. We reduce the problem to the equivalent nonlinear integral equation and we prove the complete continuity of the nonlinear integral operator. A theorem on the existence of a unique solution of the problem for the non-linear differential equation of fractional order is formulated. blasius type equation The main result of this chapter is the following theorem.

Auxiliary statements

In this section, we give some auxiliary statements for further investigation.

Theorem 5.4.1 [START_REF] Browder | A new generalization of the Schauder fixed point theorem[END_REF](Schauder fixed-point theorem.) If a completely-continuous operator A maps a bounded closed convex set K of a Banach space X into itself, then there exists at least one point x ∈ K such that Ax = x.

Consider the following operator: If m = 0, then we have 1 Γ(2α)

B t (x) = ϕ 2 (t -a) α Γ (α + 1) + ϕ 1 + ϕ 3 - ϕ 2 (b -a) α Γ (α + 1) -ϕ 1 x (t) , (5.5)
t 1 t 2 (t 1 -s) 2α-1 ds = 1 Γ(2α + 1) |t 1 -t 2 | 2α ≤ β 1 .
where 0 < β 1 = const.

If m > 0, then 1 Γ (2α) (t 1s) 2α-1 exp (-m (st 2 )) ds

= t 1 t 2 (t 1 -s) 2α-1 ∞ ∑ k=0 (-m) k (s -t 2 ) k k! ds = 1 0 (t 1 -(t 1 -t 2 ) τ -t 2 ) 2α-1 ∞ ∑ k=0 (-m) k k! ((t 1 -t 2 ) τ + t 2 -t 2 ) k (t 1 -t 2 ) dτ = ∞ ∑ k=0 (-m) k k! 1 0 (t 1 -t 2 ) 2α-1 (1 -τ) 2α-1 (t 1 -t 2 ) k τ k (t 1 -t 2 ) dτ = ∞ ∑ k=0 (-m) k k! (t 1 -t 2 ) k+2α 1 0 (1 -τ) 2α-1 τ k dτ = ∞ ∑ k=0 (-m) k k! (t 1 -t 2 ) k+2α B (2α, k + 1) = ∞ ∑ k=0 (-m) k k! (t 1 -t 2 ) k+2α Γ (2α) Γ (k + 1) Γ (k + 2α + 1) = ∞ ∑ k=0 (-m) k k! (t 1 -t 2 ) k+2α Γ (2α) k! Γ (k + 2α + 1) = Γ (2α) (t 1 -t 2 ) 2α ∞ ∑ k=0 (-m) k (t 1 -t 2 ) k Γ (k + 2α + 1) = Γ (2α) (t 1 -t 2 ) 2α E 1,2α+1 (-m (t 1 -t 2 )) then 1 Γ (2α) t 1 t 2 (t 1 -s) 2α-1 exp (-m (s -a)) ds ≤ 1 Γ (2α) t 1 t 2 (t 1 -s) 2α-1 exp (-m (s -t 2 )) ds = (t 1 -t 2 ) 2α E 1,2α+1 (-m (t 1 -t 2 )) < β 2 , 0 < β 2 = const.
Here E λ,µ (z) is a Mittag-Leffler type function. ≥ γ 2 > 0, γ 2 = const. (5.9) Inequalities (5.7) -(5.9) follow the estimate:

|B t 1 (x) -B t 2 (x)| ≤ ε,
where the positive constant ε does not depend on (x, t 1 , t 2 ) ∈ G. This ends the proof.

Proof of the main results

We reduce the problem (5.3) -(5.4) to a nonlinear integral equation. To do this, we introduce the notation D 2α * x (t) = y (t) . Then, equation (5.3) can be rewritten as y (t) + M (x (t) , t) y (t) = 0, a < t < b.

(5.10) (5.12) Next, by applying to the equation (5.12) the operator I α , and using properties 5.2.1 and 5.2.2, based on condition (5.4), we obtain

x (t) =
+ ϕ 1 + ϕ 3 - ϕ 2 (b -a) α Γ (α + 1) -ϕ 1 x (t) ≤ |ϕ 2 | (b -a) α Γ (α + 1) + |ϕ 1 | + ϕ 3 - ϕ 2 (b -a) α Γ (α + 1)
ϕ 1 ≡ R. (5.16) This means that the integral operator B t maps a ball x ≤ R into itself.

Let us prove that operator B t (x) is completely continuous. We use Arzela's theorem [START_REF] Trenogin | Functional Analysis[END_REF] on precompact sets in C ([a, b]) . (5.16) shows that image of set x ≤ R by mapping B t (x) is uniformly bounded by R. Now, we show the equicontinuity of the operator B t (x) . By Lemma 5.4.2, the operator B t (x) satisfies the condition

|B t 1 (x) -B t 2 (x)| ≤ ε.
As in the proof of Lemma 5.4.2, the positive constant ε does not depend on (x, t 1 , t 2 ) ∈ G. We have then proved that the operator B t is uniformly bounded and equicontinuous in C ([a, b]) . Consequently, by Arzelà-Ascoli' Theorem , the image B t (x) on the ball S (0, R) is pre- compact in C ([a, b]) . By consequence the operator B t is completely continuous in C ([a, b]) , and we conclude that the operator B t satisfies the Schauder's conditions in Theorem 5.4.1. Then, according to the Schauder's principle, the nonlinear integral equation (5.14) has solution in the class C ([a, b]) .

For any x (t) from the class C ([a, b]) , from the structures of operator B t in (5.5), it is easy to verify that all derivatives up to the third order of (5.5) are continuous in the weighted class C 3-α ([a, b]) . Therefore, the solution of equation (5.14) will belong to C

Conclusion

In this thesis, we first dealt with some inverse problems in fractional calculus by using Caputo derivative. Caputo's derivative allows us to impose natural initial conditions. By using the Fourier method we proved the existence and uniqueness of each solution of related inverse problems. Convergences of the obtained solutions are also justified in order to establish that the formal solutions are indeed true solutions.

Afterwards, we have examined a non-linear sequential differential (fractional analog of the Blasius equation) equation with Caputo fractional derivative. Considered problem reduced to the equivalent nonlinear integral equation and we proved the complete continuity of the nonlinear integral operator. The result is formulated on the existence of a unique solution of the problem for the non-linear differential equation of fractional order. In the limiting case, the considered boundary problem coincides with the boundary-layer problem for the Blasius equation.

I

  : B. Ahmad, A. Alsaedi, M. Kirane and Ramiz Tapdigoglu, An inverse problem for time fractional evolution equations with an involution perturbation, Quaestiones Mathematicae, 2017, vol. 40, no 2, p. 151-160. II : B.T. Torebek and Ramiz Tapdigoglu, Some inverse problems for the nonlocal heat equation with Caputo fractional derivative, June 2017, Mathematical Methods in the Applied Sciences, 2017, vol. 40, no 18, p. 6468-6479. III : Ramiz Tapdigoglu and B.T. Torebek, Inverse source problems for a wave equation with involution, Bulletin of the Karaganda University-Mathematics, 2018, vol. 91, no 3, p. 75-82. IV : Ramiz Tapdigoglu and B.T. Torebek, On a nonlinear boundarylayer problem for the fractional Blasius type equation, International Journal of Nonlinear Sciences and Numerical Simulation, 2018, vol. 19, no 5, p. 493-498.

1. 3 . 1

 31 Chapter 2: An inverse problem for a time nonlocal evolution equation with an involution perturbation . . . . . 1.3.2 Chapter 3: Some inverse problems for the nonlocal heat equation with Caputo fractional derivative . . . . . . . . 1.3.3 Chapter 4: Inverse source problems for a wave equation with involution . . . . . . . . . . . . . . . . . . . . . . . 1.3.4 Chapter 5: On a nonlinear boundary-layer problem for the fractional Blasius type equation . . . . . . . . . . . . 1.4 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Basic Functions . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Fractional Derivatives . . . . . . . . . . . . . . . . . . . . 1.4.3 Properties of Fractional Derivatives . . . . . . . . . . . .

4 1 Introduction

 1 Inverse source problems for a wave equation with involution 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Statement of problems . . . . . . . . . . . . . . . . . . . . . 4.3 Spectral properties of the perturbed Sturm-Liouville problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Proof of the uniqueness of the solution . . . . . . . . . 4.6 Proof of the existence of the solution . . . . . . . . . . . 5 On a nonlinear boundary-layer problem for the fractional blasius type equation 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Some properties of fractional operators . . . . . . . . . 5.3 Statement of the problem and main results . . . . . . . 5.4 Auxiliary statements . . . . . . . . . . . . . . . . . . . . . . 5.5 Proof of the main results . . . . . . . . . . . . . . . . . . . This chapter consists of two major parts, namely "Introduction (Française)" and "Introduction (English)Le calcul fractionnaire est une branche des mathématiques qui fait référence à l'extension du concept de dérivation classique à la dérivation d'ordre non entier. L'histoire de la dérivée d'ordre non entier remonte à la fin de 1695 quand L'Hospital a posé une question à Leibniz en s'interrogeant sur la signification de d n y dx n lorsque n = 1 2 .

  Au cours des dernières années, des avancées significatives ont été réalisées dans le domaine des problèmes inverses linéaires par Hansen [1992 a,b], [1995], Hanke et Hansen [1993], Oldenburg et al. [1991], [1993], [1994], Scales [1987], Scales et al. [1987], [1990], [1994], Parker [1994], Parker et Whaler [1981], Nolet et Snieder [1990], et d'autres.

Figure 1 . 1 -

 11 Figure 1.1 -Fractional derivatives d m dt m f (t), of a quadratic function, f (t) = t 2 (blue, solid line) with the order m which has values 0.25 (magenta, dotted line), 0.50 (black, dashed line), 0.75 (green, dash-dot line), and 1 (red, thicker solid line).

Figure 1 . 2 -

 12 Figure 1.2 -(a) The usual derivative is local, (b) The fractional derivative is nonlocal and depends on the behaviour of the entire functions.

  cant advances have been made in the field of linear inverse problems by the work of Hansen [1992 a,b], [1995], Hanke and Hansen [1993], Oldenburg et al. [1991], [1993], [1994], Scales [1987], Scales et al. [1987], [1990], [1994], Parker [1994], Parker and Whaler [1981], Nolet and Snieder [1990],

  3) has a unique solution Chapter 1. Introduction which can be written in the form u

Theorem 1 . 3 . 6

 136 If condition (*) satisfied, then problem (1.24) -(1.25) has a solution in C 3 3-α ([a, b]

g

  [START_REF] Akhmanov | Generation of structures in optical systems with two-dimensional feedback: On the way to the creation of nonlinear optical analogues of neural networks[END_REF] (x) S k (x) dx, for g = φ, ψ, ρ.

Figure 3 . 1

 31 Figure 3.1

. 16 ) 3 . 3 . 1

 16331 LemmaThe systems of functions (3.13), (3.14), (3.15),(3.16) are complete and orthonormal in L 2 (0, π) .

3. 5 .

 5 Proof of the existence of the solution of problem D 51

) 4 . 3 .

 43 Spectral properties of the perturbed Sturm-Liouville problem 57 and the homogeneous Dirichlet boundary conditions u

  ([a, b]) denotes the space:C 3 3-α ([a, b]) = x ∈ C ([a, b]) : x ∈ C 3-α ([a, b]) .HereC 3-α ([a, b]) = (ta) 3-α x ∈ C ([a, b]) .

Theorem 5 . 3 . 1

 531 If condition (*) satisfied, then problem (5.3) -(5.4) has a solution in C 3 3-α ([a, b]) .

t 1 t 2 (t 1 1 Γ

 211 s) 2α-1 exp (-m (sa)) ds ≤ (2α)

t 1 t 2 (t 1 70 Chapter 5 . 1 t 2

 2170512 s) 2α-1 exp (-m (st 2 )) ds On a nonlinear boundary-layer problem for the fractional blasius type equation where t

5. 5 .

 5 Proof of the main results 71 If m =m < 0, then 1 Γ(2α)

t 1 t 2 (t 1

 21 s) 2α-1 exp ( m(sa)) ds ≤ 1 Γ(2α)

t 1 t 2 (t 1

 21 s) 2α-1 exp ( m(st 2 )) ds = (t 1t 2 ) 2α E 1,2α+1 ( m (t 1t 2 )) < β 3 , 0 < β 3 = const.Now for the estimation ofI 2α exp -b a M (x (s) , s) ds, we consider the following cases:Case 1: Let M > 0, then taking into account the condition (*) and using formula (1.101) from[START_REF] Podlubny | Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications[END_REF], we obtainI 2α s) 2α-1 exp (-M (sa)) ds = (ba) 2α E 1,2α+1 (-M (ba)) ≥ γ 1 > 0, γ 1 = const.(5.8) Case 2: Let M = 0, then s) 2α-1 ds = (ba) 2α Γ (2α + 1)

Chapter 5 .MM

 5 On a nonlinear boundary-layer problem for the fractional blasius type equation Then, the general solution of equation (5.10) has the formy (t) = y (a) (x (s) , s) ds   , a < t < b, or equivalently, D 2α * x (t) = D 2α * x (a) (x (s) , s) ds   , a < t < b.(5.11)Applying the operator I α to both sides of the equation (5.11) and using property 5.2.2 and conditions (5.4), we getD α * x (t) = ϕ 2 + D 2α * x (a) I α , a < t < b.

ϕ 2 Γ 1 I 1 + ϕ 3 - ϕ 2 1 IM

 211321 (α + 1) (ta) α + ϕ 1 + D 2α * x (a) I 2α , a < t < b. (5.13)In (5.13), if we put t = b, and as x (b) = ϕ 3 , we haveD 2α * x (a) = ϕ 3 -ϕ 2 Γ(α+1) (ba) αϕ 2α exp -b a M (x (s) , s) ds, and the problem (5.3) -(5.4) is reduced to the following nonlinear integral equationx (t) = ϕ 2 (ta) α Γ (α + 1) + ϕ (ba) α Γ (α + 1)ϕ 2α exp -(x (s) , s) ds , a < t < b.

(5. 14 ) 5 . 5 .

 1455 Let us separately investigate operator(5.6). It is obvious that for any x ∈ C ([a, b]) when t ∈ [a, b] , we have the inequality for the operator(5.6) 0 ≤ x (t) ≤ 1. Proof of the main results73In fact, x (b) = 1, x (a) = 0 and for other case 0 < x (t) < 1. It is also easy to see that any x ∈ C ([a, b]) we have the inequality B t (x) = ϕ 2 (ta) α Γ (α + 1)

3 3 -

 3 α ([a, b]) .

  Blasius d'ordre fractionnaire. Nous le présentons au Chapitre 5.

1 Introduction 1.1 Introduction (Française) . . . . . . . . . . . . . . . . . . . 1.1.1 Calcul Fractionnaire . . . . . . . . . . . . . . . . . . . . 1.1.2 Revue historique sur les Problèmes Inverses . . . . . . . 1.1.3 Problèmes Directs . . . . . . . . . . . . . . . . . . . . . . 1.1.4 Problèmes Inverses . . . . . . . . . . . . . . . . . . . . . 1.2 Introduction (English) . . . . . . . . . . . . . . . . . . . . . 1.2.1 Fractional Calculus . . . . . . . . . . . . . . . . . . . . . 1.2.2 Historical review on Inverse Problems . . . . . . . . . . . 1.2.3 Direct Problems . . . . . . . . . . . . . . . . . . . . . . . 1.2.4 Inverse Problems . . . . . . . . . . . . . . . . . . . . . . 1.3 Presentation of the obtained results . . . . . . . . . . .

2

  An inverse problem for time fractional evolution equations with an involution perturbation 2.1 Introduction and statement of the problem . . . . . . . 2.2 The Spectral Problem . . . . . . . . . . . . . . . . . . . . . 2.3 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Proof of the Result . . . . . . . . . . . . . . . . . . . . . . . Some inverse problems for the nonlocal heat equation with Caputo fractional derivative 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Statement of problems . . . . . . . . . . . . . . . . . . . . . 3.3 Spectral properties of the perturbed Sturm-Liouville problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Proof of the existence of the solution of problem D . . 3.6 Proof of the uniqueness of the solution of problem P .

2.4.1 Existence . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Convergence of the series . . . . . . . . . . . . . . . . . . 2.4.3 Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . xi 3

Lemma 5.4.2

  Let x(t) be a continuous function on[a, b]. Then the operatorB t (x) is equicontinuity on [a, b]. Proof. For t 1 , t 2 ∈ [a, b] , t 1 > t 2 , we get |B t 1 (x) -B t 2 (x)| ≤ |ϕ 2 | Γ (α + 1) |(t 1a) α -(t 2a) α | + ϕ 3 -ϕ 2 (ba) α Γ (α + 1) ϕ 1 | x (t 1 )x (t 2 )| . (5.7)For some positive constant κ we get|ϕ 2 | Γ (α + 1) |(t 1a) α -(t 2a) α | ≤ κ.5.4. Auxiliary statements 69 For the operator x (t) we have | x (t 1 )x (t 2 )|

	I 2α exp -	t 1	M (x (s) , s) ds	-I 2α exp -
	=	a
			I 2α exp -	t	M (x (s) , s) ds
				a b	.	(5.6)
			I 2α exp -		M (x (s) , s) ds
				a

where x (t) ≡

Theorem 1.3.3 Let ϕ, ψ ∈ C 3 [0, π] and ϕ (i) (0) = ϕ (i) (π) = ψ (i) (0) = ψ (i) (π) = 0, i = 0, 1, 2.Then the solution of problem N exists, is unique and it can be written in the form

Chapter 3. Some inverse problems for the nonlocal heat equation with Caputo fractional derivative

Statement of problems

The chapter is concerned with four inverse problems concerning the time fractional heat equation with a perturbative term of involution type in the space variable. We obtain existence and uniqueness results for these problems, based on the Fourier method.

Problem D. Find the couple of functions (u (x, t) , f (x)) satisfying the equation (3.1), under the conditions u (x, 0) = ϕ (x) , x ∈ [0, π] ,

(3.2)

and the homogeneous Dirichlet boundary conditions

where ϕ (x) and ψ (x) are given sufficiently smooth functions.

Chapter 3. Some inverse problems for the nonlocal heat equation with Caputo fractional derivative Theorem 3.4.2 Let ϕ, ψ ∈ C 3 [0, π] and ϕ (i) (0) = ϕ (i) (π) = ψ (i) (0) = ψ (i) (π) = 0, i = 0, 1, 2. Then the solution of problem N exists, is unique and it can be written in the form

where

Then the solution of problem P exists, is unique and it can be written in the form

Chapter 3. Some inverse problems for the nonlocal heat equation with Caputo fractional derivative then

Similarly,

Similarly,

(

Now for the convergence of the series, we have the following estimate

Similarly for f (x) we obtain the estimate Since by hypotheses of Theorem 3.4.1, the functions ϕ (2) , ψ (2) are continuous on [0, π] , then by the Bessel inequality for the trigonometric series the following series converge:

which implies the boundedness of the set

1k , ϕ

2k , k = 1, 2, ... . Therefore, by the Weierstrass M-test (see [START_REF] Knopp | Theory of functions[END_REF]), series (3.20) and (3.21) converge absolutely and uniformly in the region Ω. Now we show the possibility of termwise differentiation of the series (3.20) twice in the variable x and once in the variable t. For this purpose, we prove that the obtained term by term differentiation of the series converge absolutely and uniformly on Ω. Given the estimates (3.22) and (3.23) we have

Hence the obtained solution satisfies the equation (3.1) point-wise; by construction, it satisfies the conditions (3.2)-(3.4).

Proof of the uniqueness of the solution of problem P

Suppose that there are two solutions {u 1 (x, t) , f 1 (x)} and {u 2 (x, t) , f 2 (x)} of problem P. Denote

Inverse source problems for a wave equation with involution

Abstract

In this chapter, a class of inverse problems for a wave equation with involution is considered for cases of two different boundary conditions, namely, Dirichlet and Neumann boundary conditions. The existence and uniqueness of solutions of these problems are proved. The solutions are obtained in the form of series expansion using a set of appropriate orthogonal basises for each problem. Convergence of the obtained solutions is also justified. Chapter 4. Inverse source problems for a wave equation with involution Similarly, the problem (4.7), (4.9) has the eigenvalues

and corresponding eigenfunctions

The following lemma is proved in [START_REF] Torebek | Some inverse problems for the nonlocal heat equation with Caputo fractional derivative[END_REF] Lemma 1 The systems of functions (4.10) and (4.11) are complete and orthonormal in L 2 (0, π) .

Main results

For the considered problems D and N, the following theorems are valid.

then the solution of problem D exists, is unique and it can be written in the form (4.13) where ψ

then the solution of problem N exists, is unique and it can be written in the form 4.5. Proof of the uniqueness of the solution Chapter 4. Inverse source problems for a wave equation with involution Substituting u 2k (t) , u 2k+1 (t) , f 2k , f 2k+1 into (4.22) and (4.23), we find

Suppose that

then

Then we have (4.12).

Similarly,

Now for the convergence of the series, we have the following estimate

Similarly for f (x) we obtain the estimate

Since by hypotheses of Theorem 1, the function ψ (4) is continuous on [0, π] , then by the Bessel inequality for the trigonometric series the following series converge: .

Therefore, by the Weierstrass M-test (see [START_REF] Knopp | Theory of functions[END_REF]), the series (4.24) and (4.25) converge absolutely and uniformly in the domain Ω. Now we show the possibility of termwise differentiation of the series (4.24) twice in the variable x and twice in the variable t. For this purpose, we prove that the series obtained by means of term by term differentiation converge absolutely and uniformly on Ω. Given the estimates (4.26) and (4.27) we have

Some properties of fractional operators

In this section, we state some basic properties of fractional differential operators.

Various properties of fractional sequential operators were studied in [START_REF] Kilbas | Theory and applications of fractional differential equations[END_REF][START_REF] Podlubny | Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications[END_REF][START_REF] Furati | Bounds on the solution of a Cauchy-type problem involving a weighted sequential fractional derivative[END_REF][START_REF] Torebek | On solvability of some boundary value problems for a fractional analogue of the Helmholtz equation[END_REF].

almost everywhere on [a, b].

Statement of the problem and main results

Consider the boundary value problem for the nonlinear fractional differential equation of Blasius type where α ∈ 1 2 , 1 and ϕ 1 , ϕ 2 , ϕ 3 are given real numbers. Note that when α = 1, problem (5.3) -(5.4) is met in boundary layer theory in fluid mechanics and polymer theory [START_REF] Glauert | The axisymmetric boundary layer on a long thin cylinder[END_REF], [START_REF] Livshitz | Hydrodynamics. Moskow[END_REF], [START_REF] Dreglea | Continuous solutions of some boundary layer problem (Poster Presentation)[END_REF], [START_REF] Guedda | Similarity solutions of differential equations for boundary layer approximations in porous media[END_REF], [START_REF] Schlichting | Boundary-Layer theory[END_REF], [START_REF] Trenogin | Functional Analysis[END_REF]. Note also that various problems for nonlinear differential equations of fractional order are investigated in [START_REF] Ahmad | On nonlocal boundary value problems for nonlinear integrodifferential equations of arbitrary fractional order[END_REF], [START_REF] Diethelm | Analysis of fractional differential equations[END_REF], [START_REF] Lakshmikantham | Basic theory of fractional differential equations[END_REF], [START_REF] Cabada | Positive solutions of nonlinear fractional differential equations with integral boundary value conditions[END_REF], [START_REF] Lin | Fractional high order methods for the nonlinear fractional ordinary differential equation[END_REF]. The recent surge in developing the theory of fractional differential equations has motivated the present work.

Condition (*)