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Heat equation

In mathematics and physics, the heat equation is a certain partial differential equation. Solutions of the heat equation are sometimes known as caloric functions. The theory of the heat equation was first developed by Joseph Fourier in 1822 for the purpose of modeling how a quantity such as heat diffuses through a given region.

As the prototypical parabolic partial differential equation, the heat equation is among the most widely studied topics in pure mathematics, and its analysis is regarded as fundamental to the broader field of partial differential equations. The heat equation can also be considered on Riemannian manifolds, leading to many geometric applications. Following work of Subbaramiah Minakshisundaram and Åke Pleijel, the heat equation is closely related with spectral geometry. A seminal nonlinear variant of the heat equation was introduced to differential geometry by James Eells and Joseph Sampson in 1964, inspiring the introduction of the Ricci flow by Richard Hamilton in 1982 and culminating in the proof of the Poincaré conjecture by Grigori Perelman in 2003. Certain solutions of the heat equation known as heat kernels provide subtle information about the region on which they are defined, as exemplified through their application to the Atiyah–Singer index theorem.

The heat equation, along with variants thereof, is also important in many fields of science and applied mathematics. In probability theory, the heat equation is connected with the study of random walks and Brownian motion via the Fokker–Planck equation. The Black–Scholes equation of financial mathematics is a small variant of the heat equation, and the Schrödinger equation of quantum mechanics can be regarded as a heat equation in imaginary time. In image analysis, the heat equation is sometimes used to resolve pixelation and to identify edges. Following Robert Richtmyer and John von Neumann's introduction of "artificial viscosity" methods, solutions of heat equations have been useful in the mathematical formulation of hydrodynamical shocks. Solutions of the heat equation have also been given much attention in the numerical analysis literature, beginning in the 1950s with work of Jim Douglas, D.W. Peaceman, and Henry Rachford Jr.

Statement of the equation
In mathematics, if given an open subset $U$ of $R^{n}$ and a subinterval $I$ of $R$, one says that a function $u : U × I → R$ is a solution of the heat equation if
 * $$\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x_1^2} + \cdots + \frac{\partial^2 u}{\partial x_n^2},$$

where $(x_{1}, …, x_{n}, t)$ denotes a general point of the domain. It is typical to refer to $t$ as "time" and $x_{1}, …, x_{n}$ as "spatial variables," even in abstract contexts where these phrases fail to have their intuitive meaning. The collection of spatial variables is often referred to simply as $x$. For any given value of $t$, the right-hand side of the equation is the Laplacian of the function $u(⋅, t) : U → R$. As such, the heat equation is often written more compactly as

In physics and engineering contexts, especially in the context of diffusion through a medium, it is more common to fix a Cartesian coordinate system and then to consider the specific case of a function $u(x, y, z, t)$ of three spatial variables $(x, y, z)$ and time variable $t$. One then says that $u$ is a solution of the heat equation if
 * $$\frac{\partial u}{\partial t} = \alpha\left(\frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}+\frac{\partial^2u}{\partial z^2}\right)$$

in which $α$ is a positive coefficient called the thermal diffusivity of the medium. In addition to other physical phenomena, this equation describes the flow of heat in a homogeneous and isotropic medium, with $u(x, y, z, t)$ being the temperature at the point $(x, y, z)$ and time $t$. If the medium is not homogeneous and isotropic, then $α$ would not be a fixed coefficient, and would instead depend on $(x, y, z)$; the equation would also have a slightly different form. In the physics and engineering literature, it is common to use $∇^{2}$ to denote the Laplacian, rather than $∆$.

In mathematics as well as in physics and engineering, it is common to use Newton's notation for time derivatives, so that $$\dot u$$ is used to denote $∂u⁄∂t$, so the equation can be written

Note also that the ability to use either $∆$ or $∇^{2}$ to denote the Laplacian, without explicit reference to the spatial variables, is a reflection of the fact that the Laplacian is independent of the choice of coordinate system. In mathematical terms, one would say that the Laplacian is "translationally and rotationally invariant." In fact, it is (loosely speaking) the simplest differential operator which has these symmetries. This can be taken as a significant (and purely mathematical) justification of the use of the Laplacian and of the heat equation in modeling any physical phenomena which are homogeneous and isotropic, of which heat diffusion is a principal example.

The "diffusivity constant" $α$ is often not present in mathematical studies of the heat equation, while its value can be very important in engineering. This is not a major difference, for the following reason. Let $u$ be a function with
 * $$\frac{\partial u}{\partial t}=\alpha\Delta u.$$

Define a new function $$v(t,x)=u(t/\alpha,x) $$. Then, according to the chain rule, one has

Thus, there is a straightforward way of translating between solutions of the heat equation with a general value of $α$ and solutions of the heat equation with $α = 1$. As such, for the sake of mathematical analysis, it is often sufficient to only consider the case $α = 1$.

Since $$\alpha>0$$ there is another option to define a $$v$$ satisfying $\frac{\partial}{\partial t} v = \Delta v $ as in ($$) above by setting $$v(t,x) = u(t, \alpha^{1/2} x) $$. Note that the two possible means of defining the new function $$v$$ discussed here amount, in physical terms, to changing the unit of measure of time or the unit of measure of length.

Physical interpretation of the equation
Informally, the Laplacian operator $∆$ gives the difference between the average value of a function in the neighborhood of a point, and its value at that point. Thus, if $$ is the temperature, $∆$ tells whether (and by how much) the material surrounding each point is hotter or colder, on the average, than the material at that point.

By the second law of thermodynamics, heat will flow from hotter bodies to adjacent colder bodies, in proportion to the difference of temperature and of the thermal conductivity of the material between them. When heat flows into (respectively, out of) a material, its temperature increases (respectively, decreases), in proportion to the amount of heat divided by the amount (mass) of material, with a proportionality factor called the specific heat capacity of the material.

By the combination of these observations, the heat equation says the rate $$\dot u$$ at which the material at a point will heat up (or cool down) is proportional to how much hotter (or cooler) the surrounding material is. The coefficient $α$ in the equation takes into account the thermal conductivity, specific heat, and density of the material.

Interpretation of the equation
The first half of the above physical thinking can be put into a mathematical form. The key is that, for any fixed $u$, one has
 * $$\begin{align}

u_{(x)}(0)&=u(x)\\ u_{(x)}'(0)&=0\\ u_{(x)}''(0)&=\frac{1}{n}\Delta u(x) \end{align}$$ where $u_{(x)}(r)$ is the single-variable function denoting the average value of $x$ over the surface of the sphere of radius $u$ centered at $r$; it can be defined by
 * $$u_{(x)}(r)=\frac{1}{\omega_{n-1}r^{n-1}}\int_{\{y:|x-y|=r\}}u\,d\mathcal{H}^{n-1},$$

in which $ω_{n − 1}$ denotes the surface area of the unit ball in $x$-dimensional Euclidean space. This formalizes the above statement that the value of $∆u$ at a point $n$ measures the difference between the value of $u(x)$ and the value of $x$ at points nearby to $u$, in the sense that the latter is encoded by the values of $u_{(x)}(r)$ for small positive values of $x$.

Following this observation, one may interpret the heat equation as imposing an infinitesimal averaging of a function. Given a solution of the heat equation, the value of $u(x, t + τ)$ for a small positive value of $τ$ may be approximated as $1⁄2n$ times the average value of the function $u(⋅, t)$ over a sphere of very small radius centered at $r$.

Character of the solutions

The heat equation implies that peaks (local maxima) of $$u$$ will be gradually eroded down, while depressions (local minima) will be filled in. The value at some point will remain stable only as long as it is equal to the average value in its immediate surroundings. In particular, if the values in a neighborhood are very close to a linear function $$A x + B y + C z + D$$, then the value at the center of that neighborhood will not be changing at that time (that is, the derivative $$\dot u$$ will be zero).

A more subtle consequence is the maximum principle, that says that the maximum value of $$u$$ in any region $$R$$ of the medium will not exceed the maximum value that previously occurred in $$R$$, unless it is on the boundary of $$R$$. That is, the maximum temperature in a region $$R$$ can increase only if heat comes in from outside $$R$$. This is a property of parabolic partial differential equations and is not difficult to prove mathematically (see below).

Another interesting property is that even if $$u$$ initially has a sharp jump (discontinuity) of value across some surface inside the medium, the jump is immediately smoothed out by a momentary, infinitesimally short but infinitely large rate of flow of heat through that surface. For example, if two isolated bodies, initially at uniform but different temperatures $$u_0$$ and $$u_1$$, are made to touch each other, the temperature at the point of contact will immediately assume some intermediate value, and a zone will develop around that point where $$u$$ will gradually vary between $$u_0$$ and $$u_1$$.

If a certain amount of heat is suddenly applied to a point in the medium, it will spread out in all directions in the form of a diffusion wave. Unlike the elastic and electromagnetic waves, the speed of a diffusion wave drops with time: as it spreads over a larger region, the temperature gradient decreases, and therefore the heat flow decreases too.

Heat flow in a uniform rod
For heat flow, the heat equation follows from the physical laws of conduction of heat and conservation of energy.

By Fourier's law for an isotropic medium, the rate of flow of heat energy per unit area through a surface is proportional to the negative temperature gradient across it:
 * $$\mathbf{q} = - k \, \nabla u $$

where $$k$$ is the thermal conductivity of the material, $$u=u(\mathbf{x},t)$$ is the temperature, and $$\mathbf{q} = \mathbf{q}(\mathbf{x},t)$$ is a vector field that represents the magnitude and direction of the heat flow at the point $$\mathbf{x}$$ of space and time $$t$$.

If the medium is a thin rod of uniform section and material, the position is a single coordinate $$x$$, the heat flow towards increasing $$x$$ is a scalar field $$q = q(t,x)$$, and the gradient is an ordinary derivative with respect to the $$x$$. The equation becomes
 * $$q = -k \,\frac{\partial u}{\partial x}$$

Let $$Q=Q(x,t)$$ be the internal heat energy per unit volume of the bar at each point and time. In the absence of heat energy generation, from external or internal sources, the rate of change in internal heat energy per unit volume in the material, $$\partial Q/\partial t$$, is proportional to the rate of change of its temperature, $$\partial u/\partial t$$. That is,
 * $$\frac{\partial Q}{\partial t} = c \, \rho \, \frac{\partial u}{\partial t}$$

where $$c$$ is the specific heat capacity (at constant pressure, in case of a gas) and $$\rho$$ is the density (mass per unit volume) of the material. This derivation assumes that the material has constant mass density and heat capacity through space as well as time.

Applying the law of conservation of energy to a small element of the medium centered at $$x$$, one concludes that the rate at which heat accumulates at a given point $$x$$ is equal to the derivative of the heat flow at that point, negated. That is,
 * $$\frac{\partial Q}{\partial t} = - \frac{\partial q}{\partial x}$$

From the above equations it follows that
 * $$\frac{\partial u}{\partial t} \;=\; - \frac{1}{c \rho} \frac{\partial q}{\partial x}

\;=\; - \frac{1}{c \rho} \frac{\partial}{\partial x} \left(-k \,\frac{\partial u}{\partial x} \right) \;=\; \frac{k}{c \rho} \frac{\partial^2 u}{\partial x^2}$$ which is the heat equation in one dimension, with diffusivity coefficient
 * $$\alpha = \frac{k}{c\rho}$$

This quantity is called the thermal diffusivity of the medium.

Accounting for radiative loss
An additional term may be introduced into the equation to account for radiative loss of heat. According to the Stefan–Boltzmann law, this term is $$\mu \left(u^4 - v^4\right)$$, where $$v=v(x,t)$$ is the temperature of the surroundings, and $$\mu$$ is a coefficient that depends on the Stefan-Boltzmann constant and the emissivity of the material. The rate of change in internal energy becomes
 * $$\frac{\partial Q}{\partial t} = - \frac{\partial q}{\partial x} - \mu \left(u^4 - v^4\right)$$

and the equation for the evolution of $$u$$ becomes
 * $$\frac{\partial u}{\partial t} = \frac{k}{c \rho} \frac{\partial^2 u}{\partial x^2} - \frac{\mu}{c \rho}\left(u^4 - v^4\right).$$

Non-uniform isotropic medium
Note that the state equation, given by the first law of thermodynamics (i.e. conservation of energy), is written in the following form (assuming no mass transfer or radiation). This form is more general and particularly useful to recognize which property (e.g. cp or $$\rho$$) influences which term.
 * $$\rho c_p \frac{\partial T}{\partial t} - \nabla \cdot \left( k \nabla T \right) = \dot q_V $$

where $$\dot q_V $$ is the volumetric heat source.

Three-dimensional problem
In the special cases of propagation of heat in an isotropic and homogeneous medium in a 3-dimensional space, this equation is
 * $$ \frac{\partial u}{\partial t} = \alpha \nabla^2 u =

\alpha \left(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2 }\right) $$   $$ = \alpha \left( u_{xx} + u_{yy} + u_{zz} \right) $$ where:
 * $$ u = u(x, y, z, t) $$ is temperature as a function of space and time;
 * $$ \tfrac{\partial u}{\partial t} $$ is the rate of change of temperature at a point over time;
 * $$ u_{xx} $$, $$ u_{yy} $$, and $$ u_{zz} $$ are the second spatial derivatives (thermal conductions) of temperature in the $$ x $$, $$ y $$, and $$ z $$ directions, respectively;
 * $$\alpha \equiv \tfrac{k}{c_p\rho}$$ is the thermal diffusivity, a material-specific quantity depending on the thermal conductivity $$ k $$, the specific heat capacity $$ c_p $$, and the mass density $$ \rho $$.

The heat equation is a consequence of Fourier's law of conduction (see heat conduction).

If the medium is not the whole space, in order to solve the heat equation uniquely we also need to specify boundary conditions for u. To determine uniqueness of solutions in the whole space it is necessary to assume additional conditions, for example an exponential bound on the growth of solutions or a sign condition (nonnegative solutions are unique by a result of David Widder).

Solutions of the heat equation are characterized by a gradual smoothing of the initial temperature distribution by the flow of heat from warmer to colder areas of an object. Generally, many different states and starting conditions will tend toward the same stable equilibrium. As a consequence, to reverse the solution and conclude something about earlier times or initial conditions from the present heat distribution is very inaccurate except over the shortest of time periods.

The heat equation is the prototypical example of a parabolic partial differential equation.

Using the Laplace operator, the heat equation can be simplified, and generalized to similar equations over spaces of arbitrary number of dimensions, as
 * $$u_t = \alpha \nabla^2 u = \alpha \Delta u, $$

where the Laplace operator, Δ or ∇2, the divergence of the gradient, is taken in the spatial variables.

The heat equation governs heat diffusion, as well as other diffusive processes, such as particle diffusion or the propagation of action potential in nerve cells. Although they are not diffusive in nature, some quantum mechanics problems are also governed by a mathematical analog of the heat equation (see below). It also can be used to model some phenomena arising in finance, like the Black–Scholes or the Ornstein-Uhlenbeck processes. The equation, and various non-linear analogues, has also been used in image analysis.

The heat equation is, technically, in violation of special relativity, because its solutions involve instantaneous propagation of a disturbance. The part of the disturbance outside the forward light cone can usually be safely neglected, but if it is necessary to develop a reasonable speed for the transmission of heat, a hyperbolic problem should be considered instead – like a partial differential equation involving a second-order time derivative. Some models of nonlinear heat conduction (which are also parabolic equations) have solutions with finite heat transmission speed.

Internal heat generation
The function u above represents temperature of a body. Alternatively, it is sometimes convenient to change units and represent u as the heat density of a medium. Since heat density is proportional to temperature in a homogeneous medium, the heat equation is still obeyed in the new units.

Suppose that a body obeys the heat equation and, in addition, generates its own heat per unit volume (e.g., in watts/litre - W/L) at a rate given by a known function q varying in space and time. Then the heat per unit volume u satisfies an equation
 * $$\frac{1}{\alpha} \frac{\partial u}{\partial t} = \left(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} \right) + \frac{1}{k}q.$$

For example, a tungsten light bulb filament generates heat, so it would have a positive nonzero value for q when turned on. While the light is turned off, the value of q for the tungsten filament would be zero.

Solving the heat equation using Fourier series
The following solution technique for the heat equation was proposed by Joseph Fourier in his treatise Théorie analytique de la chaleur, published in 1822. Consider the heat equation for one space variable. This could be used to model heat conduction in a rod. The equation is

where u = u(x, t) is a function of two variables x and t. Here
 * x is the space variable, so x ∈ [0, L], where L is the length of the rod.
 * t is the time variable, so t ≥ 0.

We assume the initial condition

where the function f is given, and the boundary conditions

Let us attempt to find a solution of $x$ that is not identically zero satisfying the boundary conditions $$ but with the following property: u is a product in which the dependence of u on x, t is separated, that is:

This solution technique is called separation of variables. Substituting u back into equation $$,
 * $$\frac{T'(t)}{\alpha T(t)} = \frac{X''(x)}{X(x)}.$$

Since the right hand side depends only on x and the left hand side only on t, both sides are equal to some constant value −λ. Thus:

and

We will now show that nontrivial solutions for $$ for values of λ ≤ 0 cannot occur:
 * 1) Suppose that λ < 0. Then there exist real numbers B, C such that $$X(x) = B e^{\sqrt{-\lambda} \, x} + C e^{-\sqrt{-\lambda} \, x}.$$ From $$ we get  X(0) = 0 = X(L) and therefore B = 0 = C which implies u is identically 0.
 * 2) Suppose that λ = 0. Then there exist real numbers B, C such that X(x) = Bx + C. From equation $$ we conclude in the same manner as in 1 that u is identically 0.
 * 3) Therefore, it must be the case that λ > 0. Then there exist real numbers A, B, C such that $$T(t) = A e^{-\lambda \alpha t}$$ and $$X(x) = B \sin\left(\sqrt{\lambda} \, x\right) + C \cos\left(\sqrt{\lambda} \, x\right).$$ From $$ we get C = 0 and that for some positive integer n, $$\sqrt{\lambda} = n \frac{\pi}{L}.$$

This solves the heat equation in the special case that the dependence of u has the special form $$.

In general, the sum of solutions to $$ that satisfy the boundary conditions $$ also satisfies $$ and $$. We can show that the solution to $$, $$ and $$ is given by
 * $$u(x,t) = \sum_{n = 1}^{\infty} D_n \sin \left(\frac{n\pi x}{L}\right) e^{-\frac{n^2 \pi^2 \alpha t}{L^2}}$$

where
 * $$D_n = \frac{2}{L} \int_0^L f(x) \sin \left(\frac{n\pi x}{L}\right ) \, dx.$$

Generalizing the solution technique
The solution technique used above can be greatly extended to many other types of equations. The idea is that the operator uxx with the zero boundary conditions can be represented in terms of its eigenfunctions. This leads naturally to one of the basic ideas of the spectral theory of linear self-adjoint operators.

Consider the linear operator Δu = uxx. The infinite sequence of functions
 * $$ e_n(x) = \sqrt{\frac{2}{L}}\sin \left(\frac{n\pi x}{L}\right)$$

for n ≥ 1 are eigenfunctions of Δ. Indeed,
 * $$ \Delta e_n = -\frac{n^2 \pi^2}{L^2} e_n. $$

Moreover, any eigenfunction f of Δ with the boundary conditions f(0) = f(L) = 0 is of the form en for some n ≥ 1. The functions en for n ≥ 1 form an orthonormal sequence with respect to a certain inner product on the space of real-valued functions on [0, L]. This means
 * $$ \langle e_n, e_m \rangle = \int_0^L e_n(x) e^*_m(x) dx = \delta_{mn}$$

Finally, the sequence {en}n ∈ N spans a dense linear subspace of L2((0, L)). This shows that in effect we have diagonalized the operator Δ.

Heat conduction in non-homogeneous anisotropic media
In general, the study of heat conduction is based on several principles. Heat flow is a form of energy flow, and as such it is meaningful to speak of the time rate of flow of heat into a region of space. q_t(V) &= - \int_{\partial V} \mathbf{H}(x) \cdot \mathbf{n}(x) \, dS \\ &= \int_{\partial V} \mathbf{A}(x) \cdot \nabla u (x) \cdot \mathbf{n}(x) \, dS \\ &= \int_V \sum_{i, j} \partial_{x_i} \bigl( a_{i j}(x) \partial_{x_j} u (x,t) \bigr)\,dx \end{align}$$
 * The time rate of heat flow into a region V is given by a time-dependent quantity qt(V). We assume q has a density Q, so that $$ q_t(V) = \int_V Q(x,t)\,d x \quad $$
 * Heat flow is a time-dependent vector function H(x) characterized as follows: the time rate of heat flowing through an infinitesimal surface element with area dS and with unit normal vector n is $$ \mathbf{H}(x) \cdot \mathbf{n}(x) \, dS .$$ Thus the rate of heat flow into V is also given by the surface integral $$ q_t(V)= - \int_{\partial V} \mathbf{H}(x) \cdot \mathbf{n}(x) \, dS $$ where n(x) is the outward pointing normal vector at x.
 * The Fourier law states that heat energy flow has the following linear dependence on the temperature gradient $$ \mathbf{H}(x) = -\mathbf{A}(x) \cdot \nabla u (x) $$ where A(x) is a 3 × 3 real matrix that is symmetric and positive definite.
 * By the divergence theorem, the previous surface integral for heat flow into V can be transformed into the volume integral $$\begin{align}
 * The time rate of temperature change at x is proportional to the heat flowing into an infinitesimal volume element, where the constant of proportionality is dependent on a constant κ $$ \partial_t u(x,t) = \kappa(x) Q(x,t)$$

Putting these equations together gives the general equation of heat flow:
 * $$ \partial_t u(x,t) = \kappa(x) \sum_{i, j} \partial_{x_i} \bigl( a_{i j}(x) \partial_{x_j} u (x,t)\bigr) $$

Remarks
 * The coefficient κ(x) is the inverse of specific heat of the substance at x × density of the substance at x: $$\kappa = 1/(\rho c_p)$$.
 * In the case of an isotropic medium, the matrix A is a scalar matrix equal to thermal conductivity k.
 * In the anisotropic case where the coefficient matrix A is not scalar and/or if it depends on x, then an explicit formula for the solution of the heat equation can seldom be written down, though it is usually possible to consider the associated abstract Cauchy problem and show that it is a well-posed problem and/or to show some qualitative properties (like preservation of positive initial data, infinite speed of propagation, convergence toward an equilibrium, smoothing properties). This is usually done by one-parameter semigroups theory: for instance, if A is a symmetric matrix, then the elliptic operator defined by $$Au(x):=\sum_{i, j} \partial_{x_i} a_{i j}(x) \partial_{x_j} u (x)$$ is self-adjoint and dissipative, thus by the spectral theorem it generates a one-parameter semigroup.

Fundamental solutions
A fundamental solution, also called a heat kernel, is a solution of the heat equation corresponding to the initial condition of an initial point source of heat at a known position. These can be used to find a general solution of the heat equation over certain domains; see, for instance, for an introductory treatment.

In one variable, the Green's function is a solution of the initial value problem (by Duhamel's principle, equivalent to the definition of Green's function as one with a delta function as solution to the first equation)
 * $$\begin{cases}

u_t(x,t) - k u_{xx}(x,t) = 0& (x, t) \in \R \times (0, \infty)\\ u(x,0)=\delta(x)& \end{cases}$$ where $$\delta$$ is the Dirac delta function. The solution to this problem is the fundamental solution (heat kernel)
 * $$\Phi(x,t)=\frac{1}{\sqrt{4\pi kt}}\exp\left(-\frac{x^2}{4kt}\right).$$

One can obtain the general solution of the one variable heat equation with initial condition u(x, 0) = g(x) for −∞ < x < ∞ and 0 < t < ∞ by applying a convolution:
 * $$u(x,t) = \int \Phi(x-y,t) g(y) dy.$$

In several spatial variables, the fundamental solution solves the analogous problem
 * $$\begin{cases}

u_t(\mathbf{x},t) - k \sum_{i=1}^nu_{x_ix_i}(\mathbf{x},t) = 0 & (\mathbf{x}, t) \in \R^n \times (0, \infty)\\ u(\mathbf{x},0)=\delta(\mathbf{x}) \end{cases}$$

The n-variable fundamental solution is the product of the fundamental solutions in each variable; i.e.,
 * $$\Phi(\mathbf{x},t) = \Phi(x_1,t) \Phi(x_2,t) \cdots \Phi(x_n,t) = \frac{1}{\sqrt{(4\pi k t)^n}} \exp \left (-\frac{\mathbf{x}\cdot\mathbf{x}}{4kt} \right).$$

The general solution of the heat equation on Rn is then obtained by a convolution, so that to solve the initial value problem with u(x, 0) = g(x), one has
 * $$u(\mathbf{x},t) = \int_{\R^n}\Phi(\mathbf{x}-\mathbf{y},t)g(\mathbf{y})d\mathbf{y}.$$

The general problem on a domain Ω in Rn is
 * $$ \begin{cases}

u_t(\mathbf{x},t) - k \sum_{i=1}^nu_{x_ix_i}(\mathbf{x},t) = 0& (\mathbf{x}, t) \in \Omega\times (0, \infty)\\ u(\mathbf{x},0)=g(\mathbf{x})&\mathbf{x}\in\Omega \end{cases}$$ with either Dirichlet or Neumann boundary data. A Green's function always exists, but unless the domain Ω can be readily decomposed into one-variable problems (see below), it may not be possible to write it down explicitly. Other methods for obtaining Green's functions include the method of images, separation of variables, and Laplace transforms (Cole, 2011).

Some Green's function solutions in 1D
A variety of elementary Green's function solutions in one-dimension are recorded here; many others are available elsewhere. In some of these, the spatial domain is (−∞,∞). In others, it is the semi-infinite interval (0,∞) with either Neumann or Dirichlet boundary conditions. One further variation is that some of these solve the inhomogeneous equation
 * $$u_{t}=ku_{xx}+f.$$

where f is some given function of x and t.

Homogeneous heat equation

 * Initial value problem on (−∞,∞) :
 * $$\begin{cases}

u_{t}=ku_{xx} & (x, t) \in \R \times (0, \infty) \\ u(x,0)=g(x) & \text{Initial condition} \end{cases} $$
 * $$u(x,t) = \frac{1}{\sqrt{4\pi kt}} \int_{-\infty}^{\infty} \exp\left(-\frac{(x-y)^2}{4kt}\right)g(y)\,dy $$

Comment. This solution is the convolution with respect to the variable x of the fundamental solution
 * $$\Phi(x,t) := \frac{1}{\sqrt{4\pi kt}} \exp\left(-\frac{x^2}{4kt}\right),$$

and the function g(x). (The Green's function number of the fundamental solution is X00.)

Therefore, according to the general properties of the convolution with respect to differentiation, u = g ∗ Φ is a solution of the same heat equation, for
 * $$\left (\partial_t-k\partial_x^2 \right )(\Phi*g)=\left [\left (\partial_t-k\partial_x^2 \right )\Phi \right ]*g=0.$$

Moreover,
 * $$\Phi(x,t)=\frac{1}{\sqrt{t}}\,\Phi\left(\frac{x}{\sqrt{t}},1\right)$$
 * $$\int_{-\infty}^{\infty}\Phi(x,t)\,dx=1,$$

so that, by general facts about approximation to the identity, Φ(⋅, t) ∗ g → g as t → 0 in various senses, according to the specific g. For instance, if g is assumed bounded and continuous on R then Φ(⋅, t) ∗ g converges uniformly to g as t → 0, meaning that u(x, t) is continuous on R × [0, ∞) with u(x, 0) = g(x).

 * Initial value problem on (0,∞) with homogeneous Dirichlet boundary conditions :
 * $$\begin{cases}

u_{t}=ku_{xx} & (x, t) \in [0, \infty) \times (0, \infty) \\ u(x,0)=g(x) & \text{IC} \\ u(0,t)=0 & \text{BC} \end{cases} $$
 * $$u(x,t)=\frac{1}{\sqrt{4\pi kt}} \int_{0}^{\infty} \left[\exp\left(-\frac{(x-y)^2}{4kt}\right)-\exp\left(-\frac{(x+y)^2}{4kt}\right)\right] g(y)\,dy $$

Comment. This solution is obtained from the preceding formula as applied to the data g(x) suitably extended to R, so as to be an odd function, that is, letting g(−x) := −g(x) for all x. Correspondingly, the solution of the initial value problem on (−∞,∞) is an odd function with respect to the variable x for all values of t, and in particular it satisfies the homogeneous Dirichlet boundary conditions u(0, t) = 0. The Green's function number of this solution is X10.

 * Initial value problem on (0,∞) with homogeneous Neumann boundary conditions :
 * $$\begin{cases}

u_{t}=ku_{xx} & (x, t) \in [0, \infty) \times (0, \infty) \\ u(x,0)=g(x) & \text{IC} \\ u_{x}(0,t)=0 & \text{BC} \end{cases} $$
 * $$u(x,t)=\frac{1}{\sqrt{4\pi kt}} \int_{0}^{\infty} \left[\exp\left(-\frac{(x-y)^2}{4kt}\right)+\exp\left(-\frac{(x+y)^2}{4kt}\right)\right]g(y)\,dy $$

Comment. This solution is obtained from the first solution formula as applied to the data g(x) suitably extended to R so as to be an even function, that is, letting g(−x) := g(x) for all x. Correspondingly, the solution of the initial value problem on R is an even function with respect to the variable x for all values of t > 0, and in particular, being smooth, it satisfies the homogeneous Neumann boundary conditions ux(0, t) = 0. The Green's function number of this solution is X20.

 * Problem on (0,∞) with homogeneous initial conditions and non-homogeneous Dirichlet boundary conditions :
 * $$\begin{cases}

u_{t}=ku_{xx} & (x, t) \in [0, \infty) \times (0, \infty) \\ u(x,0)=0 & \text{IC} \\ u(0,t)=h(t) & \text{BC} \end{cases} $$
 * $$u(x,t)=\int_{0}^{t} \frac{x}{\sqrt{4\pi k(t-s)^3}} \exp\left(-\frac{x^2}{4k(t-s)}\right)h(s)\,ds, \qquad\forall x>0$$

Comment. This solution is the convolution with respect to the variable t of
 * $$\psi(x,t):=-2k \partial_x \Phi(x,t) = \frac{x}{\sqrt{4\pi kt^3}} \exp\left(-\frac{x^2}{4kt}\right)$$

and the function h(t). Since Φ(x, t) is the fundamental solution of
 * $$\partial_t-k\partial^2_x,$$

the function ψ(x, t) is also a solution of the same heat equation, and so is u := ψ ∗ h, thanks to general properties of the convolution with respect to differentiation. Moreover,
 * $$\psi(x,t)=\frac{1}{x^2}\,\psi\left(1,\frac{t}{x^2}\right)$$
 * $$\int_0^{\infty}\psi(x,t)\,dt=1,$$

so that, by general facts about approximation to the identity, ψ(x, ⋅) ∗ h → h as x → 0 in various senses, according to the specific h. For instance, if h is assumed continuous on R with support in [0, ∞) then ψ(x, ⋅) ∗ h converges uniformly on compacta to h as x → 0, meaning that u(x, t) is continuous on [0, ∞) × [0, ∞) with u(0, t) = h(t). 2D Nonhomogeneous heat equation .gif

Inhomogeneous heat equation

 * Problem on (-∞,∞) homogeneous initial conditions :

Comment. This solution is the convolution in R2, that is with respect to both the variables x and t, of the fundamental solution
 * $$\Phi(x,t) := \frac{1}{\sqrt{4\pi kt}} \exp\left(-\frac{x^2}{4 kt}\right)$$

and the function f(x, t), both meant as defined on the whole R2 and identically 0 for all t → 0. One verifies that
 * $$\left (\partial_t-k \partial_x^2 \right )(\Phi*f)=f,$$

which expressed in the language of distributions becomes
 * $$\left (\partial_t-k \partial_x^2 \right )\Phi=\delta,$$

where the distribution δ is the Dirac's delta function, that is the evaluation at 0.

 * Problem on (0,∞) with homogeneous Dirichlet boundary conditions and initial conditions:
 * $$\begin{cases}

u_{t}=ku_{xx}+f(x,t) & (x, t) \in [0, \infty) \times (0, \infty) \\ u(x,0)=0 & \text{IC} \\ u(0,t)=0 & \text{BC} \end{cases} $$
 * $$u(x,t)=\int_{0}^{t}\int_{0}^{\infty} \frac{1}{\sqrt{4\pi k(t-s)}} \left(\exp\left(-\frac{(x-y)^2}{4k(t-s)}\right)-\exp\left(-\frac{(x+y)^2}{4k(t-s)}\right)\right)

f(y,s)\,dy\,ds $$

Comment. This solution is obtained from the preceding formula as applied to the data f(x, t) suitably extended to R × [0,∞), so as to be an odd function of the variable x, that is, letting f(−x, t) := −f(x, t) for all x and t. Correspondingly, the solution of the inhomogeneous problem on (−∞,∞) is an odd function with respect to the variable x for all values of t, and in particular it satisfies the homogeneous Dirichlet boundary conditions u(0, t) = 0.

 * Problem on (0,∞) with homogeneous Neumann boundary conditions and initial conditions :
 * $$\begin{cases}

u_{t} = ku_{xx}+f(x,t) & (x, t) \in [0, \infty) \times (0, \infty) \\ u(x,0)=0 & \text{IC} \\ u_x(0,t)=0 & \text{BC} \end{cases} $$
 * $$u(x,t)=\int_{0}^{t}\int_{0}^{\infty} \frac{1}{\sqrt{4\pi k(t-s)}} \left(\exp\left(-\frac{(x-y)^2}{4k(t-s)}\right)+\exp\left(-\frac{(x+y)^2}{4k(t-s)}\right)\right)

f(y,s)\,dy\,ds $$

Comment. This solution is obtained from the first formula as applied to the data f(x, t) suitably extended to R × [0,∞), so as to be an even function of the variable x, that is, letting f(−x, t) := f(x, t) for all x and t. Correspondingly, the solution of the inhomogeneous problem on (−∞,∞) is an even function with respect to the variable x for all values of t, and in particular, being a smooth function, it satisfies the homogeneous Neumann boundary conditions ux(0, t) = 0.

Examples
Since the heat equation is linear, solutions of other combinations of boundary conditions, inhomogeneous term, and initial conditions can be found by taking an appropriate linear combination of the above Green's function solutions.

For example, to solve
 * $$\begin{cases}

u_{t}=ku_{xx}+f & (x, t) \in \R \times (0, \infty) \\ u(x,0)=g(x) & \text{IC} \end{cases} $$ let u = w + v where w and v solve the problems
 * $$\begin{cases}

v_{t}=kv_{xx}+f, \, w_{t}=kw_{xx} \, & (x, t) \in \R \times (0, \infty) \\ v(x,0)=0,\, w(x,0)=g(x) \, & \text{IC} \end{cases} $$

Similarly, to solve
 * $$\begin{cases}

u_{t}=ku_{xx}+f & (x, t) \in [0, \infty) \times (0, \infty) \\ u(x,0)=g(x) & \text{IC} \\ u(0,t)=h(t) & \text{BC} \end{cases} $$ let u = w + v + r where w, v, and r solve the problems
 * $$\begin{cases}

v_{t}=kv_{xx}+f, \, w_{t}=kw_{xx}, \, r_{t}=kr_{xx} & (x, t) \in [0, \infty) \times (0, \infty) \\ v(x,0)=0, \; w(x,0)=g(x), \; r(x,0)=0 & \text{IC} \\ v(0,t)=0, \; w(0,t)=0, \; r(0,t)=h(t) & \text{BC} \end{cases}$$

Mean-value property for the heat equation
Solutions of the heat equations
 * $$(\partial_t -\Delta)u=0$$

satisfy a mean-value property analogous to the mean-value properties of harmonic functions, solutions of
 * $$\Delta u = 0,$$

though a bit more complicated. Precisely, if u solves
 * $$(\partial_t -\Delta)u=0$$

and
 * $$(x,t)+E_\lambda\subset\mathrm{dom}(u)$$

then
 * $$u(x,t)=\frac{\lambda}{4}\int_{E_\lambda}u(x-y,t-s)\frac{|y|^2}{s^2}ds\,dy,$$

where Eλ is a "heat-ball", that is a super-level set of the fundamental solution of the heat equation:
 * $$E_\lambda := \{(y,s) : \Phi(y,s) > \lambda\},$$
 * $$\Phi(x,t) := (4t\pi)^{-\frac{n}{2}}\exp\left(-\frac{|x|^2}{4t}\right).$$

Notice that
 * $$\mathrm{diam}(E_\lambda)=o(1)$$

as λ → ∞ so the above formula holds for any (x, t) in the (open) set dom(u) for λ large enough. This can be shown by an argument similar to the analogous one for harmonic functions.

Steady-state heat equation
The steady-state heat equation is by definition not dependent on time. In other words, it is assumed conditions exist such that:
 * $$\frac{\partial u}{\partial t} = 0$$

This condition depends on the time constant and the amount of time passed since boundary conditions have been imposed. Thus, the condition is fulfilled in situations in which the time equilibrium constant is fast enough that the more complex time-dependent heat equation can be approximated by the steady-state case. Equivalently, the steady-state condition exists for all cases in which enough time has passed that the thermal field u no longer evolves in time.

In the steady-state case, a spatial thermal gradient may (or may not) exist, but if it does, it does not change in time. This equation therefore describes the end result in all thermal problems in which a source is switched on (for example, an engine started in an automobile), and enough time has passed for all permanent temperature gradients to establish themselves in space, after which these spatial gradients no longer change in time (as again, with an automobile in which the engine has been running for long enough). The other (trivial) solution is for all spatial temperature gradients to disappear as well, in which case the temperature become uniform in space, as well.

The equation is much simpler and can help to understand better the physics of the materials without focusing on the dynamic of the heat transport process. It is widely used for simple engineering problems assuming there is equilibrium of the temperature fields and heat transport, with time.

Steady-state condition:
 * $$\frac{\partial u}{\partial t} = 0$$

The steady-state heat equation for a volume that contains a heat source (the inhomogeneous case), is the Poisson's equation:
 * $$-k \nabla^2 u = q$$

where u is the temperature, k is the thermal conductivity and q is the rate of heat generation per unit volume.

In electrostatics, this is equivalent to the case where the space under consideration contains an electrical charge.

The steady-state heat equation without a heat source within the volume (the homogeneous case) is the equation in electrostatics for a volume of free space that does not contain a charge. It is described by Laplace's equation:
 * $$\nabla^2 u = 0$$

Particle diffusion
One can model particle diffusion by an equation involving either:
 * the volumetric concentration of particles, denoted c, in the case of collective diffusion of a large number of particles, or
 * the probability density function associated with the position of a single particle, denoted P.

In either case, one uses the heat equation
 * $$c_t = D \Delta c, $$

or
 * $$P_t = D \Delta P. $$

Both c and P are functions of position and time. D is the diffusion coefficient that controls the speed of the diffusive process, and is typically expressed in meters squared over second. If the diffusion coefficient D is not constant, but depends on the concentration c (or P in the second case), then one gets the nonlinear diffusion equation.

Brownian motion
Let the stochastic process $$X$$ be the solution to the stochastic differential equation
 * $$\begin{cases}

\mathrm{d}X_t = \sqrt{2k}\; \mathrm{d}B_t \\ X_0=0 \end{cases}$$ where $$B$$ is the Wiener process (standard Brownian motion). The probability density function of $$X$$ is given at any time $$t$$ by
 * $$\frac{1}{\sqrt{4\pi kt}}\exp\left(-\frac{x^2}{4kt}\right)$$

which is the solution to the initial value problem
 * $$\begin{cases}

u_t(x,t)-ku_{xx}(x,t)=0, & (x,t)\in\R\times(0,+\infty)\\ u(x,0)=\delta(x) \end{cases}$$ where $$\delta$$ is the Dirac delta function.

Schrödinger equation for a free particle
With a simple division, the Schrödinger equation for a single particle of mass m in the absence of any applied force field can be rewritten in the following way:
 * $$\psi_t = \frac{i \hbar}{2m} \Delta \psi$$,

where i is the imaginary unit, ħ is the reduced Planck constant, and ψ is the wave function of the particle.

This equation is formally similar to the particle diffusion equation, which one obtains through the following transformation:
 * $$\begin{align}

c(\mathbf R,t) &\to \psi(\mathbf R,t) \\ D &\to \frac{i \hbar}{2m} \end{align}$$

Applying this transformation to the expressions of the Green functions determined in the case of particle diffusion yields the Green functions of the Schrödinger equation, which in turn can be used to obtain the wave function at any time through an integral on the wave function at t = 0:
 * $$\psi(\mathbf R, t) = \int \psi\left(\mathbf R^0,t=0\right) G\left(\mathbf R - \mathbf R^0,t\right) dR_x^0 \, dR_y^0 \, dR_z^0,$$

with
 * $$G(\mathbf R,t) = \left( \frac{m}{2 \pi i \hbar t} \right)^{3/2} e^{-\frac {\mathbf R^2 m}{2 i \hbar t}}.$$

Remark: this analogy between quantum mechanics and diffusion is a purely formal one. Physically, the evolution of the wave function satisfying Schrödinger's equation might have an origin other than diffusion.

Thermal diffusivity in polymers
A direct practical application of the heat equation, in conjunction with Fourier theory, in spherical coordinates, is the prediction of thermal transfer profiles and the measurement of the thermal diffusivity in polymers (Unsworth and Duarte). This dual theoretical-experimental method is applicable to rubber, various other polymeric materials of practical interest, and microfluids. These authors derived an expression for the temperature at the center of a sphere $$
 * $$\frac{T_C - T_S}{T_0 - T_S} =2 \sum_{n = 1}^{\infty} (-1)^{n+1} \exp\left({-\frac{n^2 \pi^2 \alpha t}{L^2}}\right)$$

where $T_{0}$ is the initial temperature of the sphere and $T_{C}$ the temperature at the surface of the sphere, of radius $T_{S}$. This equation has also found applications in protein energy transfer and thermal modeling in biophysics.

Applications in different areas
The heat equation arises in the modeling of a number of phenomena and is often used in financial mathematics in the modeling of options. The Black–Scholes option pricing model's differential equation can be transformed into the heat equation allowing relatively easy solutions from a familiar body of mathematics. Many of the extensions to the simple option models do not have closed form solutions and thus must be solved numerically to obtain a modeled option price. The equation describing pressure diffusion in a porous medium is identical in form with the heat equation. Diffusion problems dealing with Dirichlet, Neumann and Robin boundary conditions have closed form analytic solutions. The heat equation is also widely used in image analysis and in machine-learning as the driving theory behind scale-space or graph Laplacian methods. The heat equation can be efficiently solved numerically using the implicit Crank–Nicolson method of. This method can be extended to many of the models with no closed form solution, see for instance.

An abstract form of heat equation on manifolds provides a major approach to the Atiyah–Singer index theorem, and has led to much further work on heat equations in Riemannian geometry.

title: Heat equation


text:
Talk:Gothenberg

Unnecessary duplicate article. See Gothenburg (usual English form of the name). S.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Talk:Gothenberg


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User talk:Cat~enwiki

Hi Cat, welcome to Wikipedia. This place is highly addictive and should have a mandatory government health warning printed on the packet.

I see you have started with a small attack on that gaping empty wilderness area, Australian plants. Great! That is an area that desperately needs filling out - the number of major plant families that don't even have an entry yet is something that I've been meaning to have a go at for some time now. I hope you enjoy it here. Tannin

--

Hi Tannin! You think this place is addictive?! You should try H2G2!!

Yep, I launched a small offensive last night on the Aussie plants page - I'll hopefully add to it within the next few days, and then move on to some of the more specific plant families. I'm glad I paid attention in some of my lectures last semester...!

I really like the way this whole system works - the problem with h2g2 is that once something's in the 'edited guide', it's fairly hard to get it changed, and there's a social ladder there, on which the majority of people don't even appear. This, despite all the bells and whistles they've got over there, seems to work a lot better. Cat

Hi, I'm WD-40, if you want to know more about me, go to my page.

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Omar Bradley

Omar Nelson Bradley (February 12, 1893 –April 8, 1981) was a senior officer of the United States Army during and after World War II, rising to the rank of General of the Army. He was the first chairman of the Joint Chiefs of Staff and oversaw the U.S. military's policy-making in the Korean War.

Born in Randolph County, Missouri, he worked as a boilermaker before entering the United States Military Academy at West Point. He graduated from the academy in 1915 alongside Dwight D. Eisenhower as part of "the class the stars fell on." During World War I, he guarded copper mines in Montana. After the war, he taught at West Point and served in other roles before taking a position at the War Department under General George Marshall. In 1941, he became commander of the United States Army Infantry School.

After the U.S. entrance into World War II, he oversaw the transformation of the 82nd Infantry Division into the first American airborne division. He received his first front-line command in Operation Torch, serving under General George S. Patton in North Africa. After Patton was reassigned, Bradley commanded II Corps in the Tunisia Campaign and the Allied invasion of Sicily. He commanded the First United States Army during the Invasion of Normandy. After the breakout from Normandy, he took command of the Twelfth United States Army Group, which ultimately comprised forty-three divisions and 1.3 million men, the largest body of American soldiers ever to serve under a single field commander.

After the war, Bradley headed the Veterans Administration. He was appointed as Chief of Staff of the United States Army in 1948 and Chairman of the Joint Chiefs of Staff in 1949. In 1950, he was promoted to the rank of General of the Army, becoming the last of the nine individuals promoted to five-star rank in the United States Armed Forces. He was the senior military commander at the start of the Korean War, and supported President Harry S. Truman's wartime policy of containment. He was instrumental in persuading Truman to dismiss General Douglas MacArthur in 1951 after MacArthur resisted administration attempts to scale back the war's strategic objectives. Bradley left active duty in 1953 (although remaining on "active retirement" for the next 27 years). He continued to serve in public and business roles until his death in 1981 at age 88.

Early life and education

Omar Nelson Bradley, the son of schoolteacher John Smith Bradley (1868–1908) and his wife Mary Elizabeth (née Hubbard) (1875–1931), was born into poverty in rural Randolph County, Missouri, near Moberly. Bradley was named after Omar D. Gray, a local newspaper editor admired by his father, and a local physician, James Nelson. He was of British ancestry, his ancestors having emigrated from Great Britain to Kentucky in the mid-1700s. He attended at least eight country schools where his father taught. The elder Bradley never earned more than $40 a month in his lifetime, while he was a schoolteacher and sharecropper, the latter with the aid of all the family. They never owned a wagon, horse or a mule. When Omar was 15, his father died; he credited his father with passing on to him his love of books, baseball and shooting.

His mother moved with him to Moberly, where she remarried. Bradley graduated from Moberly High School in 1910. He was an outstanding student and athlete who was chosen captain of both the baseball and track teams.

Bradley was working as a 17-cents-an-hour boilermaker at the Wabash Railroad when he was encouraged by his Sunday school teacher at Central Christian Church in Moberly to take the entrance examination for the United States Military Academy (USMA) at West Point, New York. Bradley had been saving his money to enter the University of Missouri in Columbia, where he intended to study law. He finished second in the West Point placement exams, held at Jefferson Barracks Military Post in St. Louis, Missouri. The first-place winner was unable to accept the Congressional appointment, however, and the nomination was passed to Bradley in August 1911.

While Bradley was attending the academy, his devotion to sports prevented him from excelling academically; but he still ranked 44th in a class of 164. He was a baseball star and often played on semi-pro teams for no remuneration (to ensure his eligibility as an amateur to represent the academy). He was considered one of the most outstanding college players in the nation during his junior and senior seasons at West Point, noted as both a power hitter and an outfielder, with one of the best arms in his day. He rejected multiple offers to play professional baseball, choosing to pursue his Army career.

While stationed at West Point as an instructor, in 1923 Bradley became a Freemason. He became a member of the West Point Lodge #877, Highland Falls, New York and continued with them until his death.

Bradley married Mary Quayle (1892-1965), who had grown up across the street from him in Moberly. Her father, the town's popular police chief, had died when she was young. The pair attended Central Christian Church and Moberly High School together. On the cover of the 1910 Moberly High School yearbook, The Salutar, they were shown across from each other, although they did not date during those years. His picture bore the description "calculative" and hers "linguistic." She earned a college degree in education.

West Point and early military career

At West Point, Bradley played three years of varsity baseball including the 1914 team. Every player on that team who remained in the army ultimately became a general. Bradley graduated from West Point in 1915 as part of a class that produced many future generals, and which military historians have called "the class the stars fell on". Bradley's Cullum Number is 5356. There were ultimately 59 general officers in that graduating class, among whom Bradley and Dwight D. Eisenhower attained the rank of General of the Army. Eisenhower was elected in 1952 in a landslide victory as 34th President of the United States. Among the numerous others who became generals were Joseph T. McNarney, Henry Aurand, James Van Fleet, Stafford LeRoy Irwin, John W. Leonard, Joseph May Swing, Paul J. Mueller, Charles W. Ryder, Leland Hobbs, Vernon Prichard, John B. Wogan, Roscoe B. Woodruff, John French Conklin, Walter W. Hess, and Edwin A. Zundel.

Bradley was commissioned as a second lieutenant into the Infantry Branch of the United States Army and was first assigned to the 14th Infantry Regiment. He served on the Mexico–United States border in 1915, defending it from incursions due to the Mexican civil war. When the United States entered World War I in April 1917 (see the American entry into World War I), he was promoted to captain and sent to guard the Butte, Montana copper mines, considered of strategic importance. Bradley, who was assigned to command the second battalion of the 14th Infantry, joined the 19th Division in August 1918, which was scheduled for European deployment, but the influenza pandemic and the armistice with Germany on November 11, 1918, that fall intervened.

From September 1919 until September 1920, Bradley served as assistant professor of military science at South Dakota State College (now University) in Brookings, South Dakota.

During the difficult period between the wars, he taught and studied. From 1920 to 1924, Bradley taught mathematics at West Point. He was promoted to major in 1924 and took the advanced infantry course at Fort Benning, Georgia. After brief duty in Hawaii, Bradley was selected to study at the U.S. Army Command and General Staff School at Fort Leavenworth, Kansas in 1928–29. Upon graduating, he served as an instructor in tactics at the U.S. Army Infantry School. While Bradley was serving in this assignment, the school's assistant commandant, Lieutenant Colonel George C. Marshall, described Bradley as "quiet, unassuming, capable, with sound common sense. Absolute dependability. Give him a job and forget it."

From 1929, Bradley taught again at West Point, studying at the U.S. Army War College in 1934. Bradley was promoted to lieutenant colonel on June 26, 1936 and worked at the War Department; after 1938 he was directly reporting to U.S. Army Chief of Staff Marshall.

On February 20, 1941, Bradley was promoted to the (wartime) temporary rank of brigadier general (bypassing the rank of colonel.) (This rank was made permanent by the army in September 1943). The temporary rank was conferred to allow him to command the U.S. Army Infantry School at Fort Benning, Georgia (he was among the first from his class to reach even a temporary rank of general officer; first was his West Point classmate Luis Esteves, who was promoted Brigadier general in October 1940 ). While serving in this position he played a key part in developing the officer candidate school model.

Almost a year later, on February 15, 1942, over two months after the American entry into World War II, Bradley was made a temporary major general (a rank made permanent in September 1944) and soon took command of the 82nd Infantry Division (soon to be redesignated as the 82nd Airborne Division) before succeeding Major General James Garesche Ord as commander of the 28th Infantry Division in June.

Louisiana Maneuvers
The Louisiana Maneuvers were a series of U.S. Army exercises held around Northern and Western-Central Louisiana, including Fort Polk, Camp Claiborne and Camp Livingston, in 1940 and 1941. The exercises, which involved some 400,000 troops, were designed to evaluate U.S. training, logistics, doctrine, and commanders. Overall, headquarters were in the Bentley Hotel in Alexandria.

Many Army officers present at the maneuvers later rose to very senior roles in World War II, including Bradley, Mark Clark, Dwight Eisenhower, Walter Krueger, Lesley J. McNair and George Patton.

Lieutenant Colonel Bradley was assigned to General Headquarters during the Louisiana Maneuvers but as a courier and observer in the field, he gained invaluable experience for the future. Colonel Bradley assisted in the planning of the maneuvers, and kept the General Staff in Washington, D.C. abreast of the training that was occurring during the Louisiana Maneuvers.

Bradley later said that Louisianans welcomed the soldiers with open arms. Some soldiers even slept in some of the residents' houses. Bradley said it was so crowded in those houses sometimes when the soldiers were sleeping, there would hardly be any walking room. Bradley also said a few of the troops were disrespectful towards the residents' land and crops, and would tear down crops for extra food. However, for the most part, residents and soldiers established good relations.

World War II
Bradley's personal experiences in the war are documented in his award-winning book A Soldier's Story, published by Henry Holt & Co. in 1951. It was re-released by The Modern Library in 1999. The book is based on an extensive diary maintained by his aide-de-camp, Chester B. Hansen, who ghost-wrote the book using the diary; Hansen's original diary is maintained by the U. S. Army Heritage and Education Center, at Carlisle Barracks, Pennsylvania.

On March 25, 1942, Bradley, recently promoted to major general, assumed command of the newly activated 82nd Infantry Division. Bradley oversaw the division's transformation into the first American airborne division and took parachute training. In August the division was re-designated as the 82nd Airborne Division and Bradley relinquished command to Major General Matthew Ridgway, who had been his assistant division commander (ADC).

Bradley then took command of the 28th Infantry Division, which was a National Guard division with soldiers mostly from the state of Pennsylvania.

North Africa and Sicily
Bradley did not receive a front-line command until early 1943, after Operation Torch, the Allied invasion of French North Africa. He had been given VIII Corps after being succeeded by Lloyd D. Brown as commander of the 28th Division, but instead was sent to North Africa to be Eisenhower's front-line troubleshooter. At Bradley's suggestion, II Corps, which had just suffered a great defeat at the Kasserine Pass, was overhauled from top to bottom, and Eisenhower, now the Supreme Allied Commander of the Allied forces in North Africa, installed Major General George S. Patton as corps commander in March 1943. Patton requested Bradley as his deputy, but Bradley retained the right to represent Eisenhower as well.

Bradley succeeded Patton as commander of II Corps in April and directed it in the final Tunisian battles of April and May, with Bizerte falling to elements of II Corps on May 7, 1943. The campaign as a whole ended six days later, and with it came the surrender of over 200,000 Axis Germans and Italians.

As a result of his excellent performance in the campaign, Bradley was promoted to Brevet lieutenant general on June 2, 1943 and continued to command II Corps in the Allied invasion of Sicily (codenamed Operation Husky). The campaign lasted only a few weeks and, as he had in Tunisia, Bradley continued to impress his superiors, Eisenhower most notably, who wrote to Marshall about Bradley:

"There is very little I need to tell you about him [Bradley] because he is running absolutely true to form all the time. He has brains, a fine capacity for leadership and a thorough understanding of the requirements of modern battle. He has never caused me one moment of worry. He is perfectly capable of running an Army. He has the respect of all his associates, including all the British officers that have met him."

Normandy 1944

On September 10, 1943, Bradley transferred to London as commander in chief of the American ground forces preparing to invade France in the spring of 1944. For D-Day, Bradley was chosen to command the US First Army, which, alongside the British Second Army, commanded by Lieutenant-General Miles Dempsey, made up the 21st Army Group, commanded by General Sir Bernard Montgomery.

On June 10, 1944, four days after the initial Normandy landings, Bradley and his staff debarked to establish a headquarters ashore. During Operation Overlord, he commanded three corps directed at the two American invasion targets, Utah Beach and Omaha Beach. During July he inspected the modifications made by Curtis G. Culin to Sherman tanks, that led to the Rhino tank. Later in July, he planned Operation Cobra, the beginning of the breakout from the Normandy beachhead. Operation Cobra called for the use of strategic bombers using huge bomb loads to attack German defensive lines. After several postponements due to weather, the operation began on July 25, 1944, with a short, very intensive bombardment with lighter explosives, designed so as not to create more rubble and craters that would slow Allied progress. Bradley was horrified when 77 planes bombed short and dropped bombs on their own troops, including Lieutenant General Lesley J. McNair:

"The ground belched, shook and spewed dirt to the sky. Scores of our troops were hit, their bodies flung from slit trenches. Doughboys were dazed and frightened....A bomb landed squarely on McNair in a slit trench and threw his body sixty feet and mangled it beyond recognition except for the three stars on his collar."

However, the bombing was successful in knocking out the enemy communication system, rendering German troops confused and ineffective, and opened the way for the ground offensive by attacking infantry. Bradley sent in three infantry divisions—the 9th, 4th and 30th—to move in close behind the bombing. The infantry succeeded in cracking the German defenses, opening the way for advances by armored forces commanded by Patton to sweep around the German lines.

As the build-up continued in Normandy, the Third Army was formed under Patton, Bradley's former commander, while Lieutenant General Courtney Hodges, whom Bradley had succeeded as Commandant of the Infantry School, succeeded Bradley in command of the First Army; together, they made up Bradley's new command, the 12th Army Group. By August, the 12th Army Group had swollen to over 900,000 men and ultimately consisted of four field armies. It was the largest group of American soldiers to ever serve under one field commander.

Falaise pocket

Hitler's refusal to allow his army to flee the rapidly advancing Allied pincer movement created an opportunity to trap an entire German Army Group in northern France. After the German attempt to split the US armies at Mortain (Operation Lüttich), Bradley's Army Group and XV Corps became the southern pincer in forming the Falaise pocket, trapping the German Seventh Army and Fifth Panzer Army in Normandy. The northern pincer was formed of Canadian forces, part of British General Sir Bernard Montgomery's 21st Army Group. On August 13, 1944, concerned that American troops would clash with Canadian forces advancing from the north-west, Bradley overrode Patton's orders for a further push north towards Falaise, while ordering Major General Wade H. Haislip's XV Corps to "concentrate for operations in another direction". Any American troops in the vicinity of Argentan were ordered to withdraw. This order halted the southern pincer movement of Haislip's XV Corps. Though Patton protested the order, he obeyed it, leaving an exit—a "trap with a gap"—for the remaining German forces. Around 20,000–50,000 German troops (leaving almost all of their heavy material) escaped through the gap, avoiding encirclement and almost certain destruction. They would be reorganized and rearmed in time to slow the Allied advance into the Netherlands and Germany. Most of the blame for this outcome has been placed on Bradley. Bradley had incorrectly assumed, based on Ultra decoding transcripts, that most of the Germans had already escaped encirclement, and he feared a German counterattack as well as possible friendly fire casualties. Though admitting that a mistake had been made, Bradley placed the blame on General Montgomery for moving the British and Commonwealth troops too slowly, though the latter were in direct contact with a large number of SS Panzer, paratroopers, and other elite German forces.

Germany
The American forces reached the "Siegfried Line" or "Westwall" in late September. The success of the advance had taken the Allied high command by surprise. They had expected the German Wehrmacht to make stands on the natural defensive lines provided by the French rivers, and had not prepared the logistics for the much deeper advance of the Allied armies, so fuel ran short.

Eisenhower faced a decision on strategy. Bradley favored an advance into the Saarland, or possibly a two-thrust assault on both the Saarland and the Ruhr Area. Montgomery argued for a narrow thrust across the Lower Rhine, preferably with all Allied ground forces under his personal command as they had been in the early months of the Normandy campaign, into the open country beyond and then to the northern flank into the Ruhr, thus avoiding the Siegfried Line. Although Montgomery was not permitted to launch an offensive on the scale he had wanted, George Marshall and Hap Arnold were eager to use the First Allied Airborne Army to cross the Rhine, so Eisenhower agreed to Operation Market Garden. Bradley opposed the operation, and bitterly protested to Eisenhower the priority of supplies given to Montgomery, but Eisenhower, mindful of British public opinion regarding damage from V-1 missile launches in the north, refused to make any changes.

Bradley's Army Group now covered a very wide front in hilly country, from the Netherlands to Lorraine. Despite having the largest concentration of Allied army forces, Bradley faced difficulties in prosecuting a successful broad-front offensive in difficult country with a skilled enemy. General Bradley and his First Army commander, General Courtney Hodges, eventually decided to attack through a corridor known as the Aachen Gap towards the German township of Schmidt. The only nearby military objectives were the Roer River flood control dams, but these were not mentioned in contemporary plans and documents. Bradley and Hodges' original objective may have been to outflank German forces and prevent them from reinforcing their units further north in the Battle of Aachen. After the war, Bradley would cite the Roer dams as the objective. Since the Germans held the dams, they could also unleash millions of gallons of water into the path of advance. The campaign's confused objectives, combined with poor intelligence, resulted in the costly series of battles known as the Battle of Hurtgen Forest, which cost some 33,000 American casualties. At the end of the fighting in the Hurtgen, German forces remained in control of the Roer dams in what has been described as "the most ineptly fought series of battles of the war in the west." Further south, Patton's Third Army, which had been advancing with great speed, was faced with last priority (behind the U.S. First and Ninth Armies) for supplies, gasoline and ammunition. As a result, the Third Army lost momentum as German resistance stiffened around the extensive defenses surrounding the city of Metz. While Bradley focused on these two campaigns, the Germans were in the process of assembling troops and materiel for a surprise winter offensive.

Battle of the Bulge
Bradley's command took the initial brunt of what would become the Battle of the Bulge. For logistical and command reasons, General Eisenhower decided to place Bradley's First and Ninth Armies under the temporary command of Field Marshal Montgomery's 21st Army Group on the northern flank of the Bulge. Bradley was incensed, and began shouting at Eisenhower: "By God, Ike, I cannot be responsible to the American people if you do this. I resign." Eisenhower turned red, took a breath and replied evenly, "Brad, I—not you—am responsible to the American people. Your resignation therefore means absolutely nothing." Bradley paused, made one more protest, then fell silent as Eisenhower concluded, "Well, Brad, those are my orders."

At least one historian has attributed Eisenhower's support for Bradley's subsequent promotion to (temporary) four-star general (March 1945, not made permanent until January 1949) to, in part, a desire to compensate him for the way in which he had been sidelined during the Battle of the Bulge. Others point out that both Secretary of War Stimson and General Eisenhower had desired to reward General Patton with a fourth star for his string of accomplishments in 1944, but that Eisenhower could not promote Patton over Bradley, Devers, and other senior commanders without upsetting the chain of command (as Bradley commanded these people in the theater). A more likely explanation is that as Bradley commanded an Army Group and was the immediate subordinate of Eisenhower, who was promoted to five star rank in December 1944, it was only appropriate that he should hold the next lower rank.

Victory

Bradley used the advantage gained in March 1945—after Eisenhower authorized a difficult but successful Allied offensive (on a broad front with British Operation Veritable to the north and American Operation Grenade to the south) in February 1945—to break the German defenses and cross the Rhine into the industrial heartland of the Ruhr. Aggressive pursuit of the disintegrating German troops by the 9th Armored Division resulted in the capture of a bridge across the Rhine River at Remagen. Bradley quickly exploited the crossing, forming the southern arm of an enormous pincer movement encircling the German forces in the Ruhr from the north and south. Over 300,000 prisoners were taken. American forces then met up with the Soviet forces near the Elbe River in mid-April. By V-E Day, the 12th Army Group was a force of four armies (First, Third, Ninth, and Fifteenth) that numbered over 1.3 million men.

Command style

Unlike some of the more colorful generals of World War II, Bradley was polite and courteous in his public appearances. A reticent man, Bradley was first favorably brought to public attention by war correspondent Ernie Pyle, who was urged by General Eisenhower to "go and discover Bradley". Pyle subsequently wrote several dispatches in which he referred to Bradley as the GI's general, a title that would stay with Bradley throughout his remaining career. Will Lang Jr. of Life magazine said "The thing I most admire about Omar Bradley is his gentleness. He was never known to issue an order to anybody of any rank without saying 'Please' first."

While the public at large never forgot the image created by newspaper correspondents, a different view of Bradley was offered by combat historian S. L. A. Marshall, who knew both Bradley and George Patton, and had interviewed officers and men under their commands. Marshall, who was also a critic of George S. Patton, noted that Bradley's "common man" image "was played up by Ernie Pyle...The GIs were not impressed with him. They scarcely knew him. He's not a flamboyant figure and he didn't get out much to troops. And the idea that he was idolized by the average soldier is just rot."

While Bradley retained his reputation as the GI's general, he was criticized by some of his contemporaries for other aspects of his leadership style, sometimes described as "managerial" in nature. British General Bernard Montgomery's assessment of Bradley was that he was "dull, conscientious, dependable, and loyal". He had a habit of peremptorily relieving senior commanders who he felt were too independent, or whose command style did not agree with his own, such as the colorful and aggressive General Terry Allen, commander of the U.S. 1st Infantry Division (who was relocated to a different command because Bradley felt that his continued command of the division was making it unmanageably elitist, a decision with which Eisenhower concurred). While Patton is often viewed today as the archetype of the intolerant, impulsive commander, Bradley actually sacked far more generals and senior commanders during World War II, whereas Patton relieved only one general from his command—Orlando Ward—for cause during the entire war (and only after giving General Ward two warnings). When required, Bradley could be a hard disciplinarian; he recommended the death sentence for several soldiers while he served as the commander of the First Army.

One controversy of Bradley's leadership involved the lack of use of specialized tanks (Hobart's Funnies) in the Normandy invasion. After the war Chester Wilmot quoted correspondence with the developer of the tanks, Major General Percy Hobart, to the effect that the failure to use such tanks was a major contributing factor to the losses at Omaha Beach, and that Bradley had deferred the decision whether to use the tanks to his staff who had not taken up the offer, other than in respect of the DD (swimming) tanks. However a later memo from the 21st Army Group is on record as relaying two separate requests from the First Army, one dealing with the DD tanks and "Porpoises" (towed waterproof trailers), the other with a variety of other Funnies. The second list gives not only items of specific interest with requested numbers, but items known to be available that were not of interest. The requested items were modified Shermans, and tank attachments compatible with Shermans. Noted as not of interest were Funnies that required Churchill or Valentine tanks, or for which alternatives were available from the US. Of the six requested types of Funnies, the Sherman flamethrower version of the Churchill Crocodile is known to have been difficult to produce, and the Centipede never seems to have been used in combat. Richard Anderson considers that the press of time prevented the production of the other four items in numbers beyond the Commonwealth's requirements. Given the heavier surf and the topography of Omaha Beach, it is unlikely that the funnies would have been as useful there as they were on the Commonwealth beaches. The British had agreed to provide British-crewed Funnies to operate with the American forces but were unable to train the crews and deliver the vehicles in time.

Veterans Administration
President Truman appointed Bradley to head the Veterans Administration for two years after the war. He served from August 15, 1945, to November 30, 1947 and is credited with doing much to improve its health care system and with helping veterans receive their educational benefits under the G. I. Bill of Rights. Bradley's influence on the VA is credited with helping shape it into the agency it is today. He was a regular visitor to Capitol Hill and lobbied on behalf of veterans' benefits in testimony before various congressional veteran affairs committees. Due to his numerous contributions to the Veterans Administration, the Secretary of Veterans Affairs' primary conference room at the headquarters of the Department of Veterans Affairs is named in Bradley's honor.

Chairman of the Joint Chiefs of Staff
Bradley became the Army Chief of Staff in 1948. After assuming command, Bradley found a U.S. military establishment badly in need of reorganization, equipment, and training. As Bradley himself put it, "the Army of 1948 could not fight its way out of a paper bag."

On August 11, 1949, President Harry S. Truman appointed Bradley the first Chairman of the Joint Chiefs of Staff. After his initial 1948 plan to expand the Army and modernize its equipment was rejected by the Truman Administration, Bradley reacted to the increasingly severe postwar defense department budget cutbacks imposed by Secretary of Defense Louis A. Johnson by publicly supporting Johnson's decisions, going so far as to tell Congress that he would be doing a "disservice to the nation" if he asked for a larger military force. Bradley also suggested that official Navy protests of Secretary Johnson's canceling the supercarrier USS United States (CVA-58) were due to improper personal or political, even mutinous motives, calling Navy admirals "fancy dans who won't hit the line with all they have on every play unless they can call the signals", and who were in "open rebellion against the civilian control."

In his second memoir, Bradley would later state that not arguing more forcefully in 1948 and 1949 for a sufficient defense budget "was a mistake... perhaps the greatest mistake I made in my postwar years in Washington."

On September 22, 1950, he was promoted to the rank of General of the Army, the fifth—and last—person to achieve that rank. That same year, Bradley was made the first Chairman of the NATO Military Committee. He remained on the committee until August 1953, when he left active duty. During his service, Bradley visited the White House over 300 times and was frequently featured on the cover of Time magazine.

Bradley was also an outspoken supporter of providing aid and improving relations with Yugoslavia, stating in an address to Congress on November 30, 1950, that "In the first place, if we could even take them out of the hostile camp and make them neutral, that is one step. If you can get them to act as a threat, that's a second step. if you can get them to actively participate on your side, that is an even further step and then, of course, if you had a commitment, where their efforts were integrated with those of ours on the defence, that would still be a further step." This marked the beginning of US military aid to a communist nation in order to counter Soviet ambitions in the region, leading to greater strives in United States–Yugoslavia relations.

In 1950 Bradley was elected as an honorary member of the New York Society of the Cincinnati in recognition of his outstanding service to his country.

Korean War
As Chairman of the Joint Chiefs of Staff, Bradley was the senior military officer at the outset of the Korean War. When North Korea invaded South Korea in June 1950, Bradley was faced with re-organizing and deploying an American military force that was a shadow of its World War II counterpart. The impact of the Truman administration's defense budget cutbacks were now keenly felt, as poorly equipped American troops, lacking sufficient tanks, anti-tank weapons, or artillery were driven down the Korean peninsula to Pusan in a series of costly rearguard actions. In a postwar analysis of the unpreparedness of U.S. Army forces deployed to Korea during the summer and fall of 1950, Army Major General Floyd L. Parks stated that "Many who never lived to tell the tale had to fight the full range of ground warfare from offensive to delaying action, unit by unit, man by man...[T]hat we were able to snatch victory from the jaws of defeat...does not relieve us from the blame of having placed our own flesh and blood in such a predicament."

Bradley was the chief military policy maker during the Korean War, and supported Truman's original plan of 'rolling back' Communist aggression by conquering all of North Korea. When Chinese Communists entered North Korea in late 1950 and again drove back American forces, Bradley agreed that rollback had to be dropped in favor of a strategy of containment of North Korea. The containment strategy was subsequently adopted by the Truman administration for North Korea, and applied to communist expansion worldwide. Never an admirer of General Douglas MacArthur, Bradley was instrumental in convincing Truman to dismiss MacArthur as the overall commander in the Korean theatre after MacArthur resisted administration attempts to scale back strategic objectives in the Korean War.

In his testimony to the U.S. Congress, Bradley strongly rebuked MacArthur for his support of victory at all costs in the Korean War. Soon after Truman relieved MacArthur of command in April 1951, Bradley said in Congressional testimony, "Red China is not the powerful nation seeking to dominate the world. Frankly, in the opinion of the Joint Chiefs of Staff, this strategy would involve us in the wrong war, at the wrong place, at the wrong time, and with the wrong enemy."

Retirement

Bradley left active military service in August 1953, but remained on active duty by virtue of his rank of General of the Army. He chaired the Commission on Veterans' Pensions, commonly known as the "Bradley Commission", in 1955–1956. In January 1956, Bradley became one of the founding members of the President's Board of Consultants on Foreign Intelligence Activities, later the President's Intelligence Advisory Board.

In retirement, Bradley held a number of positions in commercial life, including Chairman of the Board of the Bulova Watch Company from 1958 to 1973. He frequently visited Moberly, Missouri, which he described as his hometown and his favorite city in the world. He was a member of the Moberly Rotary Club, regularly played near-handicap golf at the Moberly Country Club course, and had a "Bradley pew" at Central Christian Church.

His memoirs, A Soldier's Story (ghostwritten by aide-de-camp Chester B. Hansen who kept a daily diary for him during the war ), was published in 1951. Bradley started work on his autobiography A General's Life: An Autobiography (1983) before his death; it was coauthored with Clay Blair, who completed it posthumously. In this work, Bradley criticized British Field Marshal Montgomery's 1945 claims to have won the Battle of the Bulge.

On December 1, 1965, Bradley's wife, Mary, died of leukemia. He met Esther Dora "Kitty" Buhler (1922-2004) and married her on September 12, 1966; they were married until his death.

As a horse racing fan, Bradley spent much of his leisure time at racetracks in California. He was often invited to present the trophies to the winners. He was a lifetime sports fan, especially of college football. He was the 1948 Grand Marshal of the Tournament of Roses and attended several subsequent Rose Bowl games. (He was driven in his black limousine through Pasadena; it had a personalized California license plate "ONB" and a red plate with 5 gold stars. He frequently was given a police motorcycle escort to the Rose Bowl on New Year's Day.) He also was prominent at the Sun Bowl in El Paso, Texas, and the Independence Bowl in Shreveport, Louisiana in later years.

In 1967–1968 Bradley served as a member of President Lyndon Johnson's Wise Men, a high-level advisory group considering policy for the Vietnam War. Bradley was a hawk and recommended against withdrawal.

Following the death of Dwight D. Eisenhower in March 1969, Bradley was the only surviving 5-star officer in the US Armed Forces.

In 1970, Bradley served as a consultant for the film Patton. Screenwriters Francis Ford Coppola and Edmund H. North wrote most of the film based on Bradley's memoir, A Soldier's Story, and the biography, Patton: Ordeal and Triumph, by Ladislas Farago. The screenwriters did not have access to General Patton's diaries nor did Patton's family grant interviews. They relied upon observations by Bradley and other military contemporaries when attempting to reconstruct Patton's thoughts and motives.

In a review of the film Patton, S.L.A. Marshall, who knew both Patton and Bradley, stated that "The Bradley name gets heavy billing on a picture of [a] comrade that, while not caricature, is the likeness of a victorious, glory-seeking buffoon...Patton in the flesh was an enigma. He so stays in the film...Napoleon once said that the art of the general is not strategy but knowing how to mold human nature...Maybe that is all producer Frank McCarthy and Gen. Bradley, his chief advisor, are trying to say." Though each recognized that he owed part of his success to the other, it was known that Bradley disliked Patton both personally and professionally, but in the film they are portrayed as friendly.

In 1971 Bradley was the subject of an episode of the TV show This Is Your Life.

Bradley attended the 30th anniversary of D-Day at Normandy, France on June 6, 1974, participating in various parades.

On January 10, 1977, Bradley was presented with the Presidential Medal of Freedom by President Gerald Ford.

In 1978, Bradley received the Golden Plate Award of the American Academy of Achievement presented by Awards Council member General Jimmy Doolittle.

Bradley was the keynote speaker at Pointe du Hoc, Normandy, France on June 6, 1979, for the 35th anniversary of D-Day. While seated in a wheelchair, he performed an open ranks inspection of the U.S. representative army unit, the 84th Army Band from VII Corps HQ, Stuttgart, West Germany.

Bradley lived during his last years in Texas at a special residence on the grounds of the William Beaumont Army Medical Center, part of the complex which supports Fort Bliss.

One of Bradley's last public appearances was as the guest of honor at the inauguration of President Ronald Reagan on January 20, 1981.

Death
Omar Bradley died on April 8, 1981, in New York City of a cardiac arrhythmia, a few minutes after receiving an award from the National Institute of Social Sciences. He is buried at Arlington National Cemetery, next to his two wives.

Bradley served on active duty continuously from his arrival at West Point on August 1, 1911, until his death on April 8, 1981; a total of 69 years, 8 months and 7 days. His was the longest active duty career in the history of the United States Armed Forces.

Recognition and legacy
Bradley's posthumous autobiography, A General's Life, was published in 1983. Bradley began the book but found writing difficult, and hired writer Clay Blair to help shape the work. After Bradley's death, Blair continued the writing, using Bradley's first-person voice. The resulting book is also based on Blair's interviews of people in Bradley's circles, and on Bradley's personal papers.

Bradley is known for saying, "Ours is a world of nuclear giants and ethical infants. We know more about war than about peace, more about killing than we know about living."

The U.S. Army's Bradley Fighting Vehicle was named after General Bradley.

Bradley's hometown, Moberly, Missouri, classifies him as a "favorite son" and is planning a library and museum in his honor. Bradley Leadership Symposia have been held in Moberly, honoring him as a teacher of young officers. On February 12, 2010, the U.S. House of Representatives, the Missouri Senate, the Missouri House, the County of Randolph and the City of Moberly recognized Bradley's birthday as General Omar Nelson Bradley Day.

On May 5, 2000, the United States Postal Service issued a series of Distinguished Soldiers stamps in which Bradley was honored.

There is also an elementary school named after him in Fort Leavenworth, Kansas.

Assignment history

 * August 1, 1911: Cadet, United States Military Academy, West Point
 * June 12, 1915: 14th Infantry Regiment
 * September 10, 1919: ROTC professor, South Dakota State College
 * September 13, 1920: Instructor, United States Military Academy
 * September 15, 1924: Infantry School Student, Fort Benning, Georgia
 * October 1, 1925: Battalion Commander, 27th Infantry Regiment
 * June 10, 1927: Office of National Guard and Reserve Affairs, Hawaiian Department
 * August 31, 1928: Student, Command and General Staff School
 * August 1, 1929: Instructor, United States Army Infantry School, Fort Benning, Georgia
 * August 18, 1933: Student, United States Army War College
 * June 30, 1934: Plans and Training Officer, United States Military Academy
 * June 1, 1938: War Department General Staff, G-1 Chief of Operations Branch and Assistant Secretary of the General Staff
 * March 5, 1941: Commandant, U.S. Army Infantry School, Fort Benning, Georgia
 * February 19, 1942: Commanding General, 82nd Infantry Division
 * June 27, 1942: Commanding General, 28th Infantry Division
 * February 24, 1943: Personal representative in the field for Commanding General, North Africa Theater of Operations (NATO)
 * March 6, 1943: Deputy Commander, II Corps
 * April 16, 1943: Commanding General, II Corps, North Africa and Sicily
 * September 9, 1943: Commanding General, Field Forces European Theater
 * March 6, 1944: Commanding General, First Army
 * August 1, 1944: Commanding General, 12th Army Group
 * July 12, 1945: 12th Army Group disestablished, returned to United States
 * August 15, 1945: Administrator of Veterans Affairs, Veterans Administration
 * February 7, 1948: United States Army Chief of Staff
 * August 15, 1949: Chairman of the Joint Chiefs of Staff
 * August 19, 1953: Remained on active duty without an assignment

United States

 * Combat Infantry Badge.svg Combat Infantryman Badge (honorary)
 * United States Army Staff Identification Badge.png Army Staff Identification Badge
 * ArmyOSB.svg Four Overseas Service Bars

Foreign orders

 * Legion Honneur GC ribbon.svg Grand Cross, Legion of Honour (France)
 * BEL Kroonorde Grootkruis BAR.svg Grand Cross, Order of the Crown (Belgium)
 * Ordre de la couronne de Chene GC ribbon.svg Grand Cross, Order of the Oak Crown (Luxembourg)
 * GRE Order of George I - Grand Cross BAR.png Grand Cross, Order of George I (Greece)
 * GRE Order of the Phoenix - Grand Cross BAR.png Grand Cross, Order of the Phoenix (Greece)
 * Cavaliere di gran croce OMS BAR.svg Grand Cross, Military Order of Savoy (Italy)
 * Order of the Bath UK ribbon.svg Honorary Knight Commander of the Order of the Bath (United Kingdom)
 * ARG Order of the Liberator San Martin - Grand Officer BAR.png Grand Officer, Order of the Liberator (Argentina)
 * BRA Ordem do Merito Militar Grande Oficial.png Grand Officer, Order of Military Merit (Brazil)
 * Order of Orange-Nassau ribbon - Grand Officer.svg Grand Officer, Order of Orange-Nassau (Netherlands)
 * TCH Rad Bileho Lva 3 tridy (pre1990) BAR.svg Commander, Order of the White Lion (Czechoslovakia)
 * Ordre de l'Ouissam Alaouite Commandeur ribbon (Maroc).svg Commander of the Order of Ouissam Alaouite (Morocco)
 * Polonia Restituta Komandorski.jpg Commander's Cross of the Order of Polonia Restituta (Poland)
 * Order of Suvorov 106x30.png Order of Suvorov (1st class) (Soviet Union)
 * Order kutuzov1 ribbon.jpg Order of Kutuzov (1st class) (Soviet Union)

Foreign decorations and medals

 * Croix de Guerre 1939-1945 ribbon.svg UK Queen%27s Commendation for Valuable Service device.svg French Croix de guerre with silver-gilt palm
 * War Cross WWII (Belgium) with palm
 * Czechoslovak War Cross 1939–1945
 * LUX Croix de Guerre ribbon.svg Luxembourg War Cross
 * UK Queen EII Coronation Medal ribbon.svg Queen Elizabeth II Coronation Medal

Dates of rank
Source:

* – Discharged as Major and appointed Captain November 4, 1922; acts June 30, 1922, and September 14, 1922

** – Bradley's effective date for permanent brigadier general in the Regular Army is earlier than his effective date of promotion for permanent colonel. While serving as a temporary lieutenant general in early 1943, Bradley was notified that he would be promoted to permanent colonel with an effective date of October 1, 1943. At the time, promotions to permanent brigadier and major general had been withheld for more than two years, except for Delos C. Emmons, Henry H. Arnold, and Dwight Eisenhower. President Franklin D. Roosevelt lifted the moratorium after Bradley was notified that he would be promoted to colonel, but before the October 1 effective date.

In determining whom to promote after the lifting of Roosevelt's moratorium, Marshall consulted with Eisenhower, and they agreed to promote Bradley and several others. Marshall and Eisenhower then arranged the effective dates of promotion to brigadier general based on where they wanted each of the individuals selected to rank in terms of seniority. Bradley's date of rank for permanent brigadier general was then set as September 1, 1943—even though this was before his October 1, 1943, effective date for promotion to colonel—based on where Eisenhower and Marshall wanted Bradley to fall in terms of seniority as a brigadier general.

Bradley's and the other promotions to brigadier general on which Marshall and Eisenhower had conferred were not acted on until mid-October 1943 because Congress had to approve a waiver for those generals, including Bradley, who did not yet have 28 years of service. As a result, his October 1, 1943, date for promotion to permanent colonel was allowed to remain in effect. When Congress acted in mid-October to approve Bradley's time in service waiver and promotion to permanent brigadier general, his effective date for brigadier general was backdated to September 1, 1943. The September 1, 1943, date for permanent brigadier general enabled Bradley to line up with his peers where Marshall and Eisenhower intended for purposes of seniority.

The effective postdated (and then backdated) date of rank for Bradley's promotion to permanent brigadier general—September 1, 1943—thus came before the effective postdated date of rank for his promotion to colonel—October 1, 1943.

title: Omar Bradley