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User:Eml5526.s11.team4/HW5

= Problem 5.1 - G1DM1.0/D1b using WF =

Refer to lecture slide [[media:fe1.s11.mtg26.djvu|mtg-26]] for the problem statement.

Problem Statement
Solve G1DM1.0/D1b using WF with appropriate basis function (Poly, Fourier, Exp), until convergence of $$u^h(0.5)$$ to $$O(10^{-6})$$.

Boundary Conditions
The boundary conditions are

Solution
On solving the given PDE with the boundary conditions we get ,the exact equation for u(x) is as follows

Polynomial
The polynomial basis function is given by $$\mathfrak{F}_{p}=\left\{x^{j},j=0,1,...,n\right\}$$. For $$\left\{j=0,1,2,3\right\}$$ the basis functions $$\left\{b_{j}\right\}$$ are as follows: The constraint breaking solution$$\left\{\bar{b}_j\right\}$$ for the polynomial basis function is shown below: In the specific case where $$\left\{\beta=0\right\}$$ the CBS becomes:

The basis functions$$\left\{\bar{b}_j\right\}$$ satisfy the CBS requirements since$$\left\{\bar{b}_0(\beta)=\text{constant}\neq0\right\}$$ and $$\left\{\bar{b}_j(\beta)=0, for j=1,2,...,n\right\}$$.

Matlab code for polynomial function
The following figure shows the original polynomial basis functions and the CBS basis functions for $$\left\{j=1,2,3\right\}$$,We can see that convergence occurs at an n value of 8

The following figure shows the error vs n

Exponential
The exponential basis function is given by So, the basis functions $$\left\{b_j\right\}$$ are as follows: The corresponding basis functions $$\left\{\bar{b}_j\right\}$$ satisfying CBS are shown below: The basis functions$$\left\{\bar{b}_j\right\}$$ satisfy the CBS requirements

 * $$\left\{\bar{b}_0(\beta)=1\neq0\right\}$$

 * $$\left\{\bar{b}_{j}(\beta)=0, for j=1,2,...\right\}$$.

In the specific case where $$\left\{\beta=0\right\}$$ the CBS becomes:

Matlab code for exponential function
The following figure shows the original polynomial basis functions and the CBS basis functions for $$\left\{j=1,2,3\right\}$$,We can see that convergence occurs at an n value of 12

The following figure shows the error vs n

Fourier basis function

 * $$\mathfrak{F}_{f}=\left\{1, cos\frac{(2j-1)\pi x}{2}, sin(k\pi x) ,j,k=1,2,...\right\}$$

Likewise, the following figure shows the error vs n.

Plot

When n=9, the error is $$0.000004240676477$$.

The following figure is $$ u^h(x) $$

Exponential basis function

 * $$\mathfrak{F}_{e}=\left\{1, e^{i(x-1)}-1 ,i=1,2,...\right\}$$

When n = 12, $$ Error \approx 0.63\times 10^{-6} $$ and that is,

 * $$ Error\approx \begin{bmatrix}

-0.297066776970831\\ -0.114733443435346\\ 0.0165040777664232\\ 0.012641225491135\\ 0.000082171723276\\ -0.001243314977745\\ -0.000190989291357\\ 0.000100445865361\\ 0.000035683793041\\ -0.000005235050710\\ -0.000004723596919\\ -0.000000063082541 \end{bmatrix}$$

Plot
The following figure shows the error vs n. The following figure shows $$ u^h(x)$$

I will show 4 figures when n=1,2,3,4 to see how $$ u^h(x)$$ changes.

Matlab Code
= Problem 5.3 - LIBF =

Refer to lecture slide [[media:fe1.s11.mtg29.djvu|mtg-29]] for the problem statement.

Problem Statement
Repeat homework problem 5.1 by solving G1DM1.0/D1b, but use LIBF (Lagrange Interpolation Basis Functions) with uniform discretization (equidistant nodes, m = 4,6,8...). 1.) Explain how the LIBF are used as CBS (Constraint Breaking Solutions). 2.) Plot all LIBF used. 3.) Use MATLAB to integrate. 4.) Plot $$u_{m}^{h}$$ versus $$u$$ and $$u_{m}^{h}(.5)-u(.5)$$ versus $$m$$ The data set G1DM1.0/D1b is defined by an elastic bar with variable cross-section: (Image from Charles Cook, HW1.1) $$\Omega=\left\{0,1\right\}$$ $$\Gamma_{g}=\left\{0\right\}, g = 4$$ $$\Gamma _{h}=\left\{1\right\}, h = 12$$ $$a_{2}\left\{x\right\}=2+3x$$ $$f\left\{x,t\right\}=5x$$ $$\frac{\partial^{2} u}{\partial t^{2}}=0$$

Solution
Given the above information, the differential equation can be written as: $$\frac{\partial}{\partial x}\left[(2+3x)\frac{\partial u}{\partial x}\right]+5x=0$$ The general Lagrange Interpolation basis function is given by: $$L_{i,m}(x)=\prod^{m}_{k=1}\frac{x-x_k}{x_i-x_k}$$ where $$k\neq i$$ For the case where $$ m = 4 $$ the LIBF are as follows: $$L_{1,4}(x)=\frac{(x-x_{2})(x-x_{3})(x-x_{4})}{(x_{1}-x_{2})(x_{1}-x_{3})(x_{1}-x_{4})}$$ $$L_{2,4}(x)=\frac{(x-x_{1})(x-x_{3})(x-x_{4})}{(x_{2}-x_{1})(x_{2}-x_{3})(x_{2}-x_{4})}$$ $$L_{3,4}(x)=\frac{(x-x_{1})(x-x_{2})(x-x_{4})}{(x_{3}-x_{1})(x_{3}-x_{2})(x_{3}-x_{4})}$$ $$L_{4,4}(x)=\frac{(x-x_{1})(x-x_{2})(x-x_{3})}{(x_{4}-x_{1})(x_{4}-x_{2})(x_{4}-x_{3})}$$ Since the nodes are equidistant, $$x_1=0,\;x_2=\frac{1}{3},\;x_3=\frac{2}{3},\;x_4=1$$ Substituting these values into the LIBF equations and reducing produces: $$L_{1,4}(x)=\frac{(9x^{3}-18x^{2}+11x-2)}{-2}$$ $$L_{2,4}(x)=\frac{27x^{3}}{2}-\frac{45x^{2}}{2}+9x$$ $$L_{3,4}(x)=\frac{-27x^{3}}{2}+18x^{2}-\frac{9x}{2}$$ $$L_{4,1}(x)=\frac{9x^{3}}{2}-\frac{9x^{2}}{2}+x$$ This procedure was repeated for the cases of m = 6 and m = 8 using equidistant nodes. 1.) These LIBF all satisfy the CBS requirements such that, $$\mathbf{L_{1,4}(x=1)=L_{2,4}(x=1)=L_{3,4}(x=1)=0\;\; AND\;\; L_{4,4}(x=1)=1}$$ 2.) The LIBF for m = 4 are plotted below: The LIBF for m = 6 are plotted below: The LIBF for m = 8 are plotted below: 3.) The exact solution can be directly calculated by integration and applying the boundary conditions. Recall: $$\frac{\partial}{\partial x}\left[(2+3x)\frac{\partial u}{\partial x}\right]+5x=0$$ Integrating once W.R.T. 'x' produces: $$\left[(2+3x)\frac{\partial u}{\partial x}\right]+\frac{5x^{2}}{2}+C_1=0$$ Recall the natural boundary condition: $$\Gamma _{h}=\left\{1\right\}, h = 12$$ In other words: $$n(x=1)a_{2}(x+1)\frac{\partial u}{\partial x}(x=1)=12$$ Which can be applied with the results: $$\mathbf{C_1 = -14.5}$$ Which produces: $$\left[(2+3x)\frac{\partial u}{\partial x}\right]+\frac{5x^{2}}{2}-14.5=0$$ Solving the equation for 'du', integrating both sides, and applying the essential boundary condition yields: $$\mathbf{C_2 = .90651}$$ So the exact solution (strong form) can be represented by: $$u= 4.46296ln(3x+2)-\frac{5x(3x-4))}{36}+.90651$$ m=4, the exact solution and approximate solution using LIBF are compared in the following figure: The according matlab code is: m=6, the exact solution and approximate solution using LIBF are compared in the following figure: The according matlab code is:

m=8, the exact solution and approximate solution using LIBF are compared in the following figure: The according matlab code is: The error is decreased by increasing the nodes of basis function(LIBF), which can be clearly in the following figure: The according matlab code is:

Problem Statement
Continue to learn Calculix by:

a) Extract information from the “disk” example, including node information: node numbers with coordinates and element information: element number with element nodes.

b) Generate 3 meshes for the disk with triangular element with number of elements increasing.

c) Install ccx module and write a tutorial report.

a) extract information
Firstly run the Calculix command window and type the following command to open .fbd file in “build” mode.

cd disc cgx –b disc.fbd

Now we can only see the geometry, then generate mesh by the following command,

elty all qu4 mesh all plot me all

Then select: viewing->Toggle element edges you can see the shape of your elements (Fig. a-1).

Type,

plot n all

you can see the distribution of the nodes (Fig. a-2)

To see the information for a particular node or element, you may use the “ prnt ” command, such as,

Type,

prnt n 44 prnt e 30

Then you will see (Fig. a-3),

For more information about command “ prnt ” click here.

To acquire all the node and information i.e. nodal coordinates and element component, we type in the command window,

send all abq

By this action a file called “all.msh” is created and contains all nodes number and coordinates, also all element’s composition.

Opening all.msh you’ll see the following,

The node coordinates (Fig. a-4):

The element information(Fig. a-5), since we define element type “ qu4 ” i.e. “4 node shell(displayed S4 here)”, thus every element consists of 4 nodes.

b) Generate 3 meshes for the disk with triangular element with number of elements increasing
To generate new mesh, firs we need to delete the current one:

del me all

Then we create the new mesh with triangular element.

elty all tr3 mesh all plot me all

where the “ tr3 ” stands for triangular element. Alternative choices are “ tr3u ” and “ tr6 ”.

We’ll see the mesh like this (Fig b-1):

After typing “ send all abq ” and reading it, we know the number of elements of the above mesh is 384.

To generate a denser mesh i.e. with more elements, we need to change the division of lines.

Again we delete the current mesh by “ del me all ” command and then type,

plot ld all

This action will yield the following picture (Fig. b-2) telling you the division of the lines.

To get a denser mesh, we need to divide the lines into more parts. Type “ qdiv ”(with Enter) and then “ a ” (no Enter) and then type “ r ” twice (no Enter) to create a selecting window. Use the window to capture lines and change the division of them. Note that division number of 2 digits must type space in advance. Press “ q ” to quit.

Then we have change the division like (Fig b-3),

Now we can regenerate mesh,

mesh all plot me all

Remember to toggle element edge and you’ll see the denser mesh (Fig b-4), this mesh is with 864 elements.

By the same procedure we can divide the line more thoroughly and hence a very dense mesh (Fig b-5), with 1536 elements.

The preparation
For the Windows edition of Calculix, the cgx module and the ccx module are bundled together. So there’s no need to install ccx separately.

To learn the operation of ccx we can go through this tutorial to analyze a simple case: the stress/strain analysis of a disk fixed on the edge with concentrated force acting on the center.

For some reasons – more realistic, more options of element type – it’s better to replace the current 2D disk model with a 3D one. Again here we use “ del me all ” to delete mesh.

Type this command to show the label of surfaces (Fig. c-1):

plot sa all

Now we’re going to use these 4 surfaces to sweep along vertical direction to create a disk with a finite thickness. But in advance we should define them in the same set in order to manipulate them all simultaneously.

Type this command to add a new set: “set1”, with Enter button.

qadd se1

Then press “ a ” and “ r ” twice to make a selecting window (no need to press Enter). When point the window covering all 4 surfaces, press “ s ” for "surface". Finally press “ q ” to quit.

Type:

swep se1 se2 tra 0 0.25 0

This action make the original disk swep along “+y” direction making a 3D disk with 0.25 thickness (Fig. c-2).

Then we mesh de disk by,

elty all he20 mesh all plot me all

where “he20” stands for “20nodes, brick element”, and you can select your element type here.

The mesh is shown below (Fig. c-3).

To add constraint (the disk edge fixed), we need to create a set to contain nodes lying on the edge surface. First type “ plot n all ” and type “ qadd fix ” and like before press “ a ” and then “ r ” twice to select. Press “ n ” if the window is at right position. You can utilize the drop-down menu “orientation->+y direction” to help you locate. Type command “ plus n fix g ” to check whether you have chosen correctly.

When done, the cgx should display Fig. c-4.

If you mistakenly select undesired nodes you can use “ qrem ” command to remove them. The operating procedure of “ qrem ” is like “ qadd ”.

When the constraint job is over, type

send fix abq nam

to restore the selected node number into a file – “fix.nam” - for later use.

Then we can do the same procedure by adding a concentrated force in onto the center of the plate, but no need to restore them (or actually “it” since it’s a single point) into a file. We can read from the command window that it’s node #118. We can just remember this number and announce it in the ccx input file later.

Some steps before we move on to the ccx module:

 * Type “ send all abq ” creating “all.msh” file for ccx module.
 * Type “ send all fbd ” creating a .fbd file restoring geometric entities, also serve as a backup of previous work.
 * Type “ send all frd ” creating a .frd file containing node and element information.

After the all.msh file is generated, open it with WordPad to check if the element type is correct. This is because the cgx and ccx use different acronym for the same element type. And also some element type is compatiable with cgx but not accepted by ccx. This is particularly true in 2D cases. Make sure the ccx can recognize the element type appeared in all.msh. You may check it here.

The final look of the disk (Fig. c-4.5)

Running ccx
The ccx module requires an input file to run the analysis. This kind of input file is .inp format and is also called “input deck”. The input deck is generally consists of two parts: The model definition and some steps. See Fig. c-5.

And the model definition and step definition are all composed of some “keyword” parts, these the keyword with associated data is called a “card”. To help you understand these concepts see Fig. c-6.

The input deck is often written using Sci-TE text editor on Windows OS. Open an Sci-TE and paste such code onto it. And then save it with .inp suffix. This file is identical to what displayed in Fig. c-6

*Heading MODEL: DISC *INCLUDE, INPUT=all.msh *INCLUDE, INPUT=fix.nam *NSET, NSET=LOAD 118 *MATERIAL, Name=steel *ELASTIC 28000000, 0.3 *SOLID SECTION, Elset=Eall, Material=steel *STEP *STATIC *BOUNDARY Nfix,1,2,3 *CLOAD LOAD, 2, 5 *NODE FILE U *EL FILE S, E *END STEP

The heading has no effect on analysis. Let’s go to the model definition part first.

In the model definition we have,

*INCLUDE, INPUT=all.msh *INCLUDE, INPUT=fix.nam

This tells the ccx module the mesh information. The keyword “ INCLUDE ” tells ccx to read additional file which makes this file concise. You can also paste the content of additional file directly here.

*NSET, NSET=LOAD 118

This is the node where the concentrated force acts on and we name it LOAD for later use.

*MATERIAL, Name=steel *ELASTIC 28000000, 0.3 *SOLID SECTION, Elset=Eall, Material=steel

This is the Material description. “28000000” and “0.3” means Young’s Modulus and Poisson’s Ratio respectively. Since this is only a simple tutorial, we only need these parameter by assuming the material linearly elastic.

Then ahead is the step definition. Since we have the static case, only one step is needed for analysis.

*STEP *STATIC

This declare the end of model definition and the begin of step definition, it also tells the ccx module the analysis type is “static”.

*BOUNDARY Nfix,1,2,3 *CLOAD LOAD, 2, 5

This declare the boundary conditions: the edge is fixed and there’s concentrated force. Remember the set of nodes we defined before? Its name’s “fix” just as above, the following digits “123” is the direction of constraint. Keyword “ CLOAD ” stands for concentrated force(load), and “ 2 ” stands for the “y” direction while “ 5 ” stands for the magnitude.

*NODE FILE U *EL FILE S, E *END STEP

This asks the ccx for outputting data into a .frd file. U, S, E stands for data type as displacement, stress and strain respectively.

Then click on the Sci-TE toolbar -> tool -> solve or type in the command window:

ccx disc

Wait until the “Job finished!” appear.

Post Processing
Click on the Sci-TE toolbar -> post process or type in the command window “ cgx –v disc.frd ” to view the result.

The von Mises stress distribution (Fig. c-7),

The XX component of stress distribution (Fig. c-8),

The YY component of strain distribution (Fig. c-9),

As we can we the results generally reflect the fact that the load is concentrated. But the shapes of distribution look weird. This maybe caused my the unsufficiency of nodes and elements - in this 3D analysis we only allocated around 600 nodes and 100 elements. This may also be caused by my inappropriate choice of element type.

After Tutorial
For more information on the ccx module, please refer to the ccx online documentary.

And more examples available.

d) Plot figures showing comparison between exact and numerical solution, convergence vs number of element
From HW5.1/HW5.3 we have known the exact solution is,

The comparison of $$\displaystyle {{u}^{h}}$$ vs $$\displaystyle u$$:

$$\displaystyle \left[ u_{nel}^{h}(0.5)-u(0.5) \right]$$ vs number of element:

We can see that the convergence becomes better when number of elements increase.

Alternative Matlab code from Fangfang
Fangfang has developed an alternative way of coding and it's with nice outcome too, see part d).

d) Plot figures showing comparison between exact and numerical solution, convergence vs number of element
From HW5.2/HW5.4 we have known the exact solution is,

The comparison of $$\displaystyle {{u}^{h}}$$ vs $$\displaystyle u$$:

$$\displaystyle \left[ u_{nel}^{h}(0.5)-u(0.5) \right]$$ vs number of element:

We can see that the convergence becomes better when number of elements increase.

Plots by Fangfang's code
Here's 3 plots about $$\displaystyle {{u}^{h}}$$ vs $$\displaystyle u$$ when $$\displaystyle nel=4,6,8$$ generated by Fangfang's matlab code. We can it's also with nice accuracy.

title: User:Eml5526.s11.team4/HW5


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User:Egm6341.s11.team2.cleveland/HW5

=Problem 5.2: Linear State Space Model =

Given: The Linear State Space Model (LSSM)
Where LSSM has the general form
 * {| style="width:100%" border="0"

$$ \displaystyle \mathbf{x_{k+1}}_{\color{red}(nx1)} = \mathbf{F}_{\color{red}(nxn)} \mathbf{x_{k}}_{\color{red}(nx1)} $$     (2.1)
 * style="width:95%" |
 * style="width:95%" |
 *
 * }

And $$ \displaystyle \mathbf{F}_{\color{red}(nxn)} $$ is defined as
 * {| style="width:100%" border="0"

$$ \displaystyle \mathbf{F}_{\color{red}(nxn)} = \mathbf{I}_{\color{red}(nxn)} + \Delta \mathbf{A}_{\color{red}(nxn)} $$     (2.2)
 * style="width:95%" |
 * style="width:95%" |
 *
 * }

If we choose $$ \displaystyle {\color{red} n=2} $$ and define the matrices in (2.2) as
 * {| style="width:100%" border="0"

$$ \displaystyle \mathbf{I}_{\color{red}(2x2)} = \left[ \begin{matrix} 1 & 0 \\   0 & 1  \\ \end{matrix} \right], \quad \mathbf{\Delta} = 0.02, \quad \mathbf{A}_{\color{red}(2x2)} = \left[ \begin{matrix} -0.2 & 1    \\   -1   & -0.2  \\ \end{matrix} \right] $$     (2.3)
 * style="width:95%" |
 * style="width:95%" |
 *
 * }

In addition we will consider the k-th step of the system $$ \displaystyle \mathbf{x_k} $$ and initial point $$ \displaystyle \mathbf{x_0} $$, in (2.1) as
 * {| style="width:100%" border="0"

$$ \displaystyle \mathbf{x_k}_{\color{red}(2x1)} = \left\{ \begin{matrix} x_k^ \\ x_k^ \\ \end{matrix} \right\} \quad and \quad
 * style="width:95%" |
 * style="width:95%" |

\mathbf{x_0}_{\color{red}(2x1)} = \left\{ \begin{matrix} x_k^ \\ x_k^ \\ \end{matrix} \right\} = \left\{ \begin{matrix} 3 \\   -2  \\ \end{matrix} \right\} $$     (2.3)
 *
 * }

1. Run LSSM
And plot $$ \displaystyle \{ \mathbf{x_j}, j=0,1,...\} $$ in the state space $$ \displaystyle \left( x^1,x^2\right) =\left( x,y\right) $$

2. Find the Equilibrium Point
As $$ \displaystyle \underset{x \to \infty}{\mathop{\lim }} \mathbf{x_{k+1}} = \underset{x \to \infty}{\mathop{\lim }} \mathbf{F^{k+1}\cdot x_0} =: \mathbf{ \hat{x}} $$

a) Plot $$ \mathbf{ \hat{x}} $$
Using a BIG RED DOT

b) Plot $$ \displaystyle \mathbf{x_} $$
Using a BIG BLUE DOT in the same plane as $$ \displaystyle \{ \mathbf{x_j}, j=0,1,...\} $$ using small dots

3. Gaussian Random Noise:
====a) Let $$ \displaystyle \mathbf{G} = \alpha \cdot \left\{ \begin{matrix}  1  \\   1  \\ \end{matrix} \right\}_{\color{red}2x1}  $$.====

4. Cauchy Random Noise:
====a) Let $$ \displaystyle \mathbf{G} = \alpha \cdot \left\{ \begin{matrix}  1  \\   1  \\ \end{matrix} \right\}_{\color{red}2x1}  $$.====

Hint:
Find a Matlab command to generate $$ \displaystyle \{ \mathbf{\theta_{j}}, j=0,1,2,... \} $$ in single-slit diffraction experiment.

1. Run LSSM
And plot $$ \displaystyle \{ \mathbf{x_j}, j=0,1,...\} $$ in the state space $$ \displaystyle \left( x^1,x^2\right) =\left( x,y\right) $$

Matlab Code: Matlab Plot of the LSSM:
 * HW5-P2-1.tif

Using the above Matlab code we were able to track the time evolution over time to the equilibrium point. We considered $$ \displaystyle n = 1001 $$ points to obtain the equilibrium position. It is reasonable to infer that this behavior continues as $$ \displaystyle n \rightarrow \infty $$ since the stability of the phase space by eigenvalue analysis holds. Namely, for a given $$ \displaystyle 2x2 $$ matrix $$ \displaystyle \mathbf{A} $$ of the general form,

 * {| style="width:100%" border="0"

$$ \displaystyle \mathbf{A}_{\color{red}(2x2)} = \left[ \begin{matrix} a & b \\ c & d \\ \end{matrix} \right] $$     (2.4)
 * style="width:95%" |
 * style="width:95%" |
 *
 * }

Remarks:
Must satisfy the following two conditions to be classified as stable:

i) The trace, $$ \displaystyle T = a+d < 0 $$ and

ii) The determinant, $$ \displaystyle D = ad-bc > 0 $$

For our matrix $$ \displaystyle \mathbf{A} $$ we get:

i) The trace, $$ \displaystyle T = -0.2+(-0.2) = -0.4 < 0 $$ and

ii) The determinant, $$ \displaystyle D = (-0.2)(-0.2)-(-1)(1)=1.04 > 0 $$

Therefore, we observe a stable near-equilibrium at $$ \displaystyle n=1001 $$ which sufficiently demonstrates the behavior of this system. Since it was easy enough to carry out the numerical verification at larger $$ \displaystyle $$ a calculation for $$ \displaystyle n=10001 $$ which provided points around the order of $$ \displaystyle 10^{-17} $$.

2. Find the Equilibrium Point
As $$ \displaystyle k $$ goes to infinity,

a) Plot $$ \mathbf{ \hat{x}} $$
Using a BIG RED DOT

b) Plot $$ \displaystyle \mathbf{x_} $$
Using a BIG BLUE DOT in the same plane as $$ \displaystyle \{ \mathbf{x_j}, j=0,1,...\} $$ using small dots

Matlab Code:

Matlab Plot of the LSSM:
 * HW5-P2-1.tif

3. Gaussian Random Noise:
====a) Let $$ \displaystyle \mathbf{G} = \alpha \cdot \left\{ \begin{matrix}  1  \\   1  \\ \end{matrix} \right\}_{\color{red}2x1}  $$.====

c) Using the Matlab command randn to generate $$ \displaystyle \{ \mathbf{w_{j}}, j=0,1,2,... \} $$.
Matlab Code:

d) Plot $$ \displaystyle \{ \mathbf{x_{j}}, j=0,1,2,... \} $$ for $$ \displaystyle \alpha=0.5,1,2 $$.
Shown Below is the Linear State Space Model With Random Gaussian Noise and $$ \displaystyle {\color{red}\alpha = 0.5} $$:
 * HW5-P2-3-guassianNoiseAlpha_05.tif

Shown Below is the Linear State Space Model With Random Gaussian Noise and $$ \displaystyle {\color{red}\alpha = 1} $$:
 * HW5-P2-3-guassianNoiseAlpha_2.tif

Shown Below is the Linear State Space Model With Random Gaussian Noise and $$ \displaystyle {\color{red}\alpha = 2} $$:
 * HW5-P2-3-guassianNoiseAlpha_2.tif

4. Cauchy Random Noise:
====a) Let $$ \displaystyle \mathbf{G} = \alpha \cdot \left\{ \begin{matrix}  1  \\   1  \\ \end{matrix} \right\}_{\color{red}2x1}  $$.====

c) Using the Matlab command randn to generate $$ \displaystyle \{ \mathbf{w_{j}}, j=0,1,2,... \} $$.
Matlab Code:

d) Plot $$ \displaystyle \{ \mathbf{x_{j}}, j=0,1,2,... \} $$ for $$ \displaystyle \alpha=0.5,1,2 $$.
Shown Below is the Linear State Space Model With Cauchy Noise and $$ \displaystyle {\color{red}\alpha = 0.5} $$:
 * HW5-P2-4cauchyNoiseAlpha_05.tif

Shown Below is the Linear State Space Model With Cauchy Noise and $$ \displaystyle {\color{red}\alpha = 1} $$:
 * HW5-P2-4-cauchyNoiseAlpha_1.tif

Shown Below is the Linear State Space Model With Cauchy Noise and $$ \displaystyle {\color{red}\alpha = 2} $$:
 * HW5-P2-4cauchyNoiseAlpha_2.tif

Hint:
Find a Matlab command to generate $$ \displaystyle \{ \mathbf{\theta_{j}}, j=0,1,2,... \} $$ in single-slit diffraction experiment.

=Problem 5.7: =

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

title: User:Egm6341.s11.team2.cleveland/HW5


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Innovation and Entrepreneurship/Tools

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title: Innovation and Entrepreneurship/Tools


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Monash University page on blogs tells everything about using blogs in teaching. This is an excellent page with links to popular blogging services, educational blogs and case studies on the use of blogs in teaching different subjects.

2. Social media in teaching (In Finnish)– LeMill

LeMill is a web community for finding, authoring and sharing learning resources.Social media in teaching by Teemu Leinonen and  Social media and network sites for teachers by Riina Vuorikari  provide descriptions and examples of use for different types of social media tools for different teaching needs.

3. Twitter in classroom

31 interesting ways to use Twitter in classroom, a presentation by Doug Belshaw gives simple ways to use Twitter for information retrieval, connecting to people etc,.


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Innovation and Entrepreneurship/Links

 1. Social Media group in Cloudworks

This cloudworks link connects to exemplary groups collaborating to learn and share ideas and experiences in using the Web2.0 tools, social media tools and e-learning technologies. A must see site for educators interested in Web 2.0 tools for teaching.

2. Educator 2.0

Educator 2.0 is a free social knowledge network for everyone who uses the web to educate people. In this group innovative educators share their knowledge with one another. Classes are held in live online classrooms and can be created by everyone in the community. This is an excellent group to teach and study about the newest technologies and techniques in online education.

title: Innovation and Entrepreneurship/Links


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Innovation and Entrepreneurship/Case Examples

 1. Second Life to teach Entrepreneurship

An interesting article on using Second Life to teach entrepreneurship can be found in this link. The article is by Howard Van Auken, Iowa State University .

Also, follow this link to view a Slideshare presentation by Patrycja Rudnicka about using Second Life to teach Entrepreneurship.

2. Going SOLO

Going SOLO is a curriculum designed by the University of Illinois Extension staff that includes activities that encourage creative problem solving, critical thinking, and communication skills.In addition to the activities, Going SOLO uses an online simulation game to enhance learning.Going Solo, an article by Darlene Knipe et al., discusses the use of the simulation game to teach entrepreneurship.

3. The Mean Business of Second Life

`The Mean Business of Second Life: Teaching Entrepreneurship, Technology and e-Commerce in Immersive Environments` an article in the MERLOT Journal of Online Learning and Teaching (JOLT) describes a Masters of Business Administration elective course, in which Second Life was utilized to teach strategic and managerial issues related to e-commerce. The article is authored by Brian Mennecke et.al

 '''Please create case examples in this section to show the application of Web 2.0 tools and Social Media tools in teaching Innovation and Entrepreneurship. Feel free to share your creative ways of using the web tools for the benefit of educators around the globe'''


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Open web sources for teaching innovation & entrepreneurship

 1. TALCIE

TALCIE stands for Teaching and Learning for Creativity, Innovation and Entrepreneurship. The TALCIE community of practice is a group of people - students, alumni, teachers and practitioners - that elicits the participation of its members and community to make it easier for us to enable the best possible learning experience for people who want to become more creative, innovative and entrepreneurial. TALCIE offers free educational resources under Creative Commons license on various creative and innovative entrepreneurial activities and competitions. Venture Design Studio under the The Entrepreneurial Experience Course by Dr.Alex Bruton provides the method to conduct a Venture pitching and assessing session. The Mount Royal Innovation Tournament contains interesting videos of students’ competition entries on creating and measuring as much value as possible from ordinary objects like a business card, string, paper bag, plastic bottle etc.,

2. The Stanford Technology Ventures Program (STVP)

The Stanford Technology Ventures Program (STVP) is the entrepreneurship education and research center located within the School of Engineering at Stanford University.The Stanford Technology Ventures Program (STVP) Entrepreneurship Corner is a free online archive of entrepreneurship resources for teaching and learning. STVP offers a wide range of teaching resources in the form of links to books, journals, videos etc.,

STVP's Entrepreneurship Corner is an internationally recognized resource for educators who teach students about technology and high-growth entrepreneurship. It is an online archive of content from the Stanford Technology Ventures Program and leading entrepreneurship faculty at universities world-wide. This constantly-expanding resource includes an extensive searcheable database of video clips and podcasts of entrepreneurial thought leaders; reading lists; case studies on entrepreneurship; and links to conferences, foundations, and groups that support entrepreneurship education.

3. MIT Sloan Teaching Innovation Resources

MIT Sloan Teaching Innovation Resources (MSTIR) is a collection of teaching materials, including case studies, simulations, deep dives, and industry, business and country overviews that MIT Sloan provides as a free teaching resource open and available to the world. The various materials posted on this site have been developed by MIT Sloan faculty and students. The areas currently include the global financial crisis, sustainability, industry evolution, and global entrepreneurship.

MIT OpenCourseWare (OCW) is a web-based publication of virtually all Massachusetts Institute of Technology course content. OCW is open and available to the world and is a permanent MIT activity. It is a free publication of MIT course materials that reflects almost all the undergraduate and graduate subjects taught at MIT. Excellent teaching resources are available on Managing Innovation and Entrepreneurship and Developmental Entrepreneurship. A search function with key words is an efficient way to find the resources in this website.


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Open Web Sources for Teaching Innovation & Entrepreneurship- Links to online interest groups/Social Media groups/Angel Investor groups for entrepreneurs

 1. London Women Mean Business – LinkedIn group

London Women Mean Business is a networking group managed by Business Link in London. Use the group to network, access free business support, find out about free events, and ask questions about all aspects of running a small or medium sized business in London. Exclusive invites to women-only events and access to free guides and information.

2. UnLtdWorld – Social Network for Entrepreneurs

UnLtdWorld is a open platform social network aimed at connecting social entrepreneurs, social innovators and socially-minded people. It allows entrepreneurs to build a personal business community, make connections, collaborate and network to make new contacts. It also helps to promote business, get business insights and find resources, toolkits and guides.


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Open Web Sources for Teaching Innovation & Entrepreneurship- Links to entrepreneurial websites with useful information

 1. International Council for Small Business (ICSB)

The International Council for Small Business (ICSB) was the first international membership organization to promote the growth and development of small businesses worldwide. It is devoted to the advancement of management development practices for potential entrepreneurs and existing small business owner/managers through education, research, and the free exchange of ideas. The knowledge center offers information on all of ICSB conference proceedings, white papers, innovative teaching resources, and experiential learning materials for registered members. The knowledge links to SME and Entrepenuership Blogs and a number of SME business websites.

2. StartupNation

StartupNation is a free service founded by entrepreneurs for entrepreneurs. It has grown to be the leading online content and community resource for entrepreneurs. It connects to the local business and entrepreneurial networks in the North America and contains a lot of useful information for budding entrepreneurs. Links to podcasts, articles and business advise blogs can also be found here.

3. Business Link in London - Social Enterprise

Business Link in London for Social enterprise is a free, impartial, independent business advisory service available to anyone starting up or running a small or medium sized business. BLL provides high quality, actionable information and practical advice, connecting to expert assistance to help one to succeed. BLL website contains excellent free tool kit for setting up social enterprises, guides on business plans to market analysis, videos and many more useful resources for entrepreneurs. The Resource Directory is a valuable repository of case studies, templates, tips and fact sheets on a wide array of entrepreneurial topics. The other BLL links are, [www.businesslink.gov.uk/london/women Women's Online Centre for female entrepreneurs] and [www.businesslink.gov.uk/london/enable ENABLE website ] for deaf and disabled entrepreneurs.


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Open Web Sources for Teaching Innovation & Entrepreneurship- Links to Articles/Publications/Videos/Books

 1. BBC’s Dragons’ Den

BBC’s Dragons’Den is a series of video shows on Entrepreneurs pitching for investment in the Den from the ‘ Dragons’, who are  five venture capitalists willing to invest their own money into the ventures of the budding entrepreneurs, in  exchange for equity. The Dragons Den webpage also contain an online resource bank called the ‘Business School’ to learn about nine essential elements of business. It also connects to many of BBC business links and external sources useful for doing business.


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Innovation and Entrepreneurship/Open Educational Resources

 Links to open educational web resources for teaching and learning in many fields are listed here.

1. Wikieducator Open Educational Resources

Wikieducator portal providing links to exemplary open educational resources, institutional repositories, subject portals and collections and community developed content in many fields of education. This is a valuable portal for educators around the globe.

2. The Aspen Institute Center for Business Education (CBE)

The Aspen Institute Center for Business Education equips business leaders for the 21st century with a new management paradigm—the vision and knowledge to integrate corporate profitability and social value. CBE is a part of the Aspen Institute Business and Society Program (BSP), an organization dedicated to developing leaders for a sustainable global society.

Caseplace.org is a dynamic and practical on-line resource for up-to-date case studies, syllabi, and innovative teaching materials on business and sustainability - from corporate governance and ethical issues to sustainable development. It is the leading resource for business school teaching materials that incorporate social and environmental elements into mainstream b-school discussion topics.CasePlace.org partners with case publishers and top business journals to connect the real-world practice of sustainability to education in the business school classroom. Their case collection is carefully selected from the largest and most prominent case publishers, including Harvard Business School Publishing, ecch, Darden and Ivey and the most high quality materials that touch carefully on social, environmental and ethical issues are chosen.

3.Cloudworks

The vision statement of Cloudworks’ states that "it is a site for finding, sharing and discussing learning and teaching ideas, experiences and issues”. Cloudworks  emphasis is on building a dynamic collection of ideas and experiences; via a variety of educational content (learning designs, case studies, resources and tools) plus active discussions about the use and effectiveness of this content in different contexts.

4.Open Course Ware Consortium

Many links to OCW courses and OCW organization at OCW Consortium


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Innovation and Entrepreneurship/Open Web Sources

 1. TALCIE

TALCIE stands for Teaching and Learning for Creativity, Innovation and Entrepreneurship. The TALCIE community of practice is a group of people - students, alumni, teachers and practitioners - that elicits the participation of its members and community to make it easier for us to enable the best possible learning experience for people who want to become more creative, innovative and entrepreneurial. TALCIE offers free educational resources under Creative Commons license on various creative and innovative entrepreneurial activities and competitions. Venture Design Studio under the The Entrepreneurial Experience Course by Dr.Alex Bruton provides the method to conduct a Venture pitching and assessing session. The Mount Royal Innovation Tournament contains interesting videos of students’ competition entries on creating and measuring as much value as possible from ordinary objects like a business card, string, paper bag, plastic bottle etc.,

2. The Stanford Technology Ventures Program (STVP)

The Stanford Technology Ventures Program (STVP) is the entrepreneurship education and research center located within the School of Engineering at Stanford University.The Stanford Technology Ventures Program (STVP) Entrepreneurship Corner is a free online archive of entrepreneurship resources for teaching and learning. STVP offers a wide range of teaching resources in the form of links to books, journals, videos etc.,

STVP's Entrepreneurship Corner is an internationally recognized resource for educators who teach students about technology and high-growth entrepreneurship. It is an online archive of content from the Stanford Technology Ventures Program and leading entrepreneurship faculty at universities world-wide. This constantly-expanding resource includes an extensive searcheable database of video clips and podcasts of entrepreneurial thought leaders; reading lists; case studies on entrepreneurship; and links to conferences, foundations, and groups that support entrepreneurship education.

3. MIT Sloan Teaching Innovation Resources

MIT Sloan Teaching Innovation Resources (MSTIR) is a collection of teaching materials, including case studies, simulations, deep dives, and industry, business and country overviews that MIT Sloan provides as a free teaching resource open and available to the world. The various materials posted on this site have been developed by MIT Sloan faculty and students. The areas currently include the global financial crisis, sustainability, industry evolution, and global entrepreneurship.

MIT OpenCourseWare (OCW) is a web-based publication of virtually all Massachusetts Institute of Technology course content. OCW is open and available to the world and is a permanent MIT activity. It is a free publication of MIT course materials that reflects almost all the undergraduate and graduate subjects taught at MIT. Excellent teaching resources are available on Managing Innovation and Entrepreneurship and Developmental Entrepreneurship. A search function with key words is an efficient way to find the resources in this website.


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Innovation and Entrepreneurship/Online Interest Groups

 1. London Women Mean Business – LinkedIn group

London Women Mean Business is a networking group managed by Business Link in London. Use the group to network, access free business support, find out about free events, and ask questions about all aspects of running a small or medium sized business in London. Exclusive invites to women-only events and access to free guides and information.

2. UnLtdWorld – Social Network for Entrepreneurs

UnLtdWorld is a open platform social network aimed at connecting social entrepreneurs, social innovators and socially-minded people. It allows entrepreneurs to build a personal business community, make connections, collaborate and network to make new contacts. It also helps to promote business, get business insights and find resources, toolkits and guides.

3. European Entrepreneurship & Innovation @ Silicon Valley

This LinkedIn and Facebook group serves Europeans, Americans and other entrepreneurs, venture investors, universities, corporations, technology transfer offices, innovation professionals, service providers, academics and policy makers with an interest in European technology entrepreneurship and innovation, practised in Silicon Valley, northern California and/or Europe. Europe includes those countries from Ireland to Russia, and from Scandinavia to the Mediterranean.


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Innovation and Entrepreneurship/Entrepreneurial Websites

 1. International Council for Small Business (ICSB)

The International Council for Small Business (ICSB) was the first international membership organization to promote the growth and development of small businesses worldwide. It is devoted to the advancement of management development practices for potential entrepreneurs and existing small business owner/managers through education, research, and the free exchange of ideas. The knowledge center offers information on all of ICSB conference proceedings, white papers, innovative teaching resources, and experiential learning materials for registered members. The knowledge links to SME and Entrepenuership Blogs and a number of SME business websites.

2. StartupNation

StartupNation is a free service founded by entrepreneurs for entrepreneurs. It has grown to be the leading online content and community resource for entrepreneurs. It connects to the local business and entrepreneurial networks in the North America and contains a lot of useful information for budding entrepreneurs. Links to podcasts, articles and business advise blogs can also be found here.

3. Business Link in London - Social Enterprise

Business Link in London for Social enterprise is a free, impartial, independent business advisory service available to anyone starting up or running a small or medium sized business. BLL provides high quality, actionable information and practical advice, connecting to expert assistance to help one to succeed. BLL website contains excellent free tool kit for setting up social enterprises, guides on business plans to market analysis, videos and many more useful resources for entrepreneurs. The Resource Directory is a valuable repository of case studies, templates, tips and fact sheets on a wide array of entrepreneurial topics. The other BLL links are, [www.businesslink.gov.uk/london/women Women's Online Centre for female entrepreneurs] and [www.businesslink.gov.uk/london/enable ENABLE website ] for deaf and disabled entrepreneurs.

4. European Entrepreneurship and Innovation at Stanford

Stanford Engineering's European Entrepreneurship and Innovation Thought Leaders (ME421) is a weekly speaker series that presents industry leaders from Europe's hitech startup, venture finance, corporate and university research and technology commercialization communities to share their insights and experiences with aspiring and veteran entrepreneurs from Silicon Valley. The website contains information on Entrepreneurship European programmes, Roundtables on Entrepreneurship Education (REE),  Business, Professional and Cultural Networking Organizations in Northern California, list of recommended readings and slide share presentations of the ME421 speakers.

5. US Small Business Administration (SBA)

SBA provides support to small businesses and small-business owners across the USA. The website contains excellent resources to get one started with the pre-steps necessary to start a business. There are links to learn about  business plan, legal obligations, financing options and many more. An important and valuable link for the educators and entrepreneurs alike.


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Innovation and Entrepreneurship/Videos

 1. BBC’s Dragons’ Den

BBC’s Dragons’Den is a series of video shows on Entrepreneurs pitching for investment in the Den from the ‘ Dragons’, who are  five venture capitalists willing to invest their own money into the ventures of the budding entrepreneurs, in  exchange for equity. The Dragons Den webpage also contain an online resource bank called the ‘Business School’ to learn about nine essential elements of business. It also connects to many of BBC business links and external sources useful for doing business.

title: Innovation and Entrepreneurship/Videos


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Innovation and Entrepreneurship/Creative Games

 1. Wisc-Online Game Builder

Wisc-Online is a digital library of Web-based learning resources called "learning objects." The digital library of objects has been developed primarily by faculty from the Wisconsin Technical College System (WTCS) and produced by multimedia technicians who create the learning objects. Wisc-Online offers an ever-growing library of high-quality educational teaching materials free to educators.

Wisc-Online offers a creative tool called game builder. The users can choose from templates for many different games and enter their content in a game format of their choice: Baseball, Build Your Fortune, Matching, Tic-Tac-Toe, Jeopardy, Case Study, Hangman, LearningLand, Quiz, Game Show, Bingo, TimeOut, Spin to Win, and Media Quiz. These online 'make your own games' are used for educating students in a fun way. A free registration is needed to avail this service.

2. Going SOLO

Going SOLO emphasizes the whole range of skills required for success. Going SOLO is a curriculum designed by the University of Illinois Extension staff that includes activities that encourage creative problem solving, critical thinking, and communication skills. In addition to the activities, Going SOLO uses an online simulation game to enhance learning.

3. Gamestorming

Gamestorming is a very creative blog that teaches about designing games that help produce innovative, creative results in everyone’s work. The blog contains archives of games and information of how to play them and also how to design them to fit one’s own specific work goals. The blog also has the video reviews of the book ‘Gamestorming: A Playbook for Innovators, Rulebreakers, and Changemakers’ by Dave Gray, Sunni Brown and James Macanufo.


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Innovation and Entrepreneurship/Experts

 '''This section connects the users of this portal to the extremely creative educators in the field of Innovation and Entrepreneurship. Their contributions on the creative techniques that they have designed and recommend is shared here for educational and non-commercial use. Educators all around the globe are welcome to be a part of this creative group and encouraged to share their creative teaching techniques, recommended readings, favorite educational games etc, in this section.'''

Dr Peter Kelly
Senior Lecturer, School of Science, Aalto University MBA (University of Notre Dame USA), PhD (London Business School UK)

Peter Kelly is an award winning entrepreneurship educator and practitioner. He obtained is PhD in Entrepreneurship from London Business School (UK) and an MBA from the University of Notre Dame (USA). Over the past 15 years, he has worked with entrepreneurs in Finland, Sweden, UK, Portugal, Russia, Latvia, Estonia, Norway, Switzerland, and Canada. Since arriving in Finland in 1998, Peter has been the driving force behind entrepreneurship education at Aalto University heading up the Helsinki School of Creative Entrepreneurship, an initiative that involved all the Schools and the largest private donor to Aalto University. During 2010, he held an appointment as Professor of Practice at the School of Art & Design where he developed a rapid opportunity prototyping course to explore the intersection between the disciplines of design and entrepreneurship. Dr Kelly was directly involved in the creation of Europe’s first university pre-seed fund and an associated angel network. He is also a business angel.

Peter Kelly's Creative Techniques
Peter Kelly’s creative exercises focus on generating ideas, team building, creating novelty, and handling surprise by exploring how to make something memorable from a fixed set of inputs in a fun way.

[[Media:Effectual_Coffee.pdf|Effectual Coffee By Peter Kelly]]

[[Media:Effectual Drinks.pdf|Effectual Drinks By Peter Kelly]]

[[Media:Effectual Meal.pdf|Effectual Meal By Peter Kelly]]

[[Media:Idea Workshop Instructions.pdf|Idea Workshop By Peter Kelly]]

[[Media:Mobile Creativity Instructions.pdf|Mobile Creativity By Peter Kelly]]

title: Innovation and Entrepreneurship/Experts


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User:Pandaparts

Polyglot with an arts background who wants to learn to program. Error?

title: User:Pandaparts


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Communication and Identities in Institutional Arenas - Part I/Wenger, Etienne (1998). Communities of Practice. Learning, Meaning and Identity.

Summary and reading log of Wenger, Etienne, 1998: Communities of Practice – learning, meaning and identity
'''N.B.! This reading log is incomplete!'''

The book presents a theory of learning that is based on an assumption of social practices being fundamental for human beings’ learning and identity development. Learning is thus seen as a process of social participation, and the primary unit of analysis “communities of practice” that people form in different contexts. These communities of practice are everywhere and we all participate in several communities of practice at any given time. The term may be new, but the experience is not, Wenger states (p.7).

Introduction – A social theory of learning

Point of departure for the book is a critical perspective of what usually has been considered as learning – an individual, separate process resulting from teaching. The challenge is thus described as the following: to place learning in the context of “our lived experience of participation in the world” (p.3)

Four premises that Wenger states in terms of learning: 1.	Human beings are social beings. 2.	Knowledge is a matter of competence with respect to valued enterprises 3.	Knowing is a matter of participating in the pursuit of such enterprises – or active engagement in the world 4.	Meaning – our ability to experience the world and our engagement with it as meaningful – is ultimately what learning is to produce. (p.4)

Participation for Wenger is being active participants of social communities and constructing identities in relation to these communities. Participation shapes what we do, who we are and how we interpret what we do.

Wenger also presents a model of four components of a social theory of learning (p.5):

 * Learning as belonging - community

 * Learning as doing – practice
 * Learning as becoming – identity

 * Learning as experience - meaning

The different components are defined as follows: Meaning: a way of talking about our (changing) ability to experience our lives and the world as meaningful Practice: a way of talking about the shared historical resources, frameworks and perspectives that can sustain mutual engagement in action Community: a way of talking about the social configurations in which our enterprises are defined as worth pursuing and our participation is recognizable as competence Identity: a way of talking about how learning changes who we are and creates personal histories of becoming in the context of our communities

About schools (p.6): CoPs emerge everywhere, in the classroom as well as on the playground, officially and inofficially, in spite of curriculum and because of it. According to Wenger, the learning that is most personally transformative is the kind of learning that involves membership in these CoPs. He therefore suggests that we rethink and revise our understanding of learning. The theory of social learning proposed by Wenger is built at the intersection of two main axes of intersecting traditions. On the vertical axis, the theories of social structure and situated experience and on the horizontal axis the theories of practice and identity (p.12). The social theory of learning, in the intersection of these, implies that learning as participation takes place through our engagement in actions and interactions, embedded in culture and history. Learning is also “the vehicle for the evolution of practices… as well as for the development and transformation of identities” (p.13). Wenger reminds us also of the relevance of theories of collectivity – subjectivity, power and meaning (p.14).

Short summary of the different parts of the book:

Part I Practice provides as series of characterizations of the concept of CoP, including the following:
 * The level of analysis at which the concept of practice is useful
 * The defining characteristics of a CoP
 * The evolution of CoPs over time
 * Boundaries and relations among CoPs
 * Constellations formed by interrelated CoPs
 * An essay: “Knowing in practice”

Part II focuses on identity with the same analytical perspective and...
 * Injects the notion of the person into the theory without having to posit an individual subject to start with
 * Expands the domain of inquiry to social configurations – contexts for identity formations
 * Requires a theory of power by which to characterize the formation of identity in practice as the ability to negotiate an experience of meaning

In the epilogue then, Wenger discusses two kinds of social design, of which I will focus on education and the formation of identities later in this text.


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User:Eml5526.s11.team7.jin/HW5

=5.1=

Exact solution
The governing partial differential equation (PDE):
 * {| style="width:100%" border="0"

$$ \displaystyle
 * style="width:95%" |
 * style="width:95%" |

\frac{\partial }\left[ {\left( {2 + 3x} \right)\frac} \right] + 5x = 0{\text{ }}\forall {\text{x}} \in \left] {0,1} \right[

$$     (Eq ) The natural and essential B.C.:
 *
 * }
 * {| style="width:100%" border="0"

$$ \displaystyle
 * style="width:95%" |
 * style="width:95%" |

\frac\left( 1 \right) = \frac{5}

$$     (Eq )
 *
 * }
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$$ \displaystyle
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u\left( 0 \right) = 4

$$     (Eq ) Integrating
 *
 * }
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$$ \displaystyle
 * style="width:95%" |
 * style="width:95%" |

\int {\frac{\partial }\left[ {\left( {2 + 3x} \right)\frac} \right]} dx = \int { - 5xdx}

$$     (Eq )
 *
 * }
 * {| style="width:100%" border="0"

$$ \displaystyle
 * style="width:95%" |
 * style="width:95%" |

\left( {2 + 3x} \right)\frac + {C_1} = - \frac{5}{2}{x^2}{\text{ where }}{C_1}{\text{ is a constant}}

$$     (Eq ) By the natural B.C.
 *
 * }
 * {| style="width:100%" border="0"

$$ \displaystyle
 * style="width:95%" |
 * style="width:95%" |

{C_1} = - \frac{2}

$$     (Eq ) Then
 *
 * }
 * {| style="width:100%" border="0"

$$ \displaystyle
 * style="width:95%" |
 * style="width:95%" |

\frac = \frac

$$     (Eq )
 *
 * }
 * {| style="width:100%" border="0"

$$ \displaystyle
 * style="width:95%" |
 * style="width:95%" |

u = - \frac{5}{x^2} + \frac{5}{9}x + \frac\log \left( {3x + 2} \right) + {C_2}{\text{ where }}{C_2}{\text{ is a constant}}

$$     (Eq ) By the essential B.C.
 *
 * }
 * {| style="width:100%" border="0"

$$ \displaystyle
 * style="width:95%" |
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{C_2} = 4 - \frac\log \left( 2 \right)

$$     (Eq ) Then
 *
 * }
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$$ \displaystyle
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * style="width:92%; padding:10px; border:2px solid #8888aa" |

u = - \frac{5}{x^2} + \frac{5}{9}x + \frac\log \left( {3x + 2} \right) + 4 - \frac\log \left( 2 \right)

$$     (Eq )
 *
 * }

=5.2=

Exact solution
The governing partial differential equation (PDE):
 * {| style="width:100%" border="0"

$$ \displaystyle
 * style="width:95%" |
 * style="width:95%" |

\frac{\partial }\left[ {\left( {2 + 3x} \right)\frac} \right] + 5x = 0{\text{ }}\forall {\text{x}} \in \left] {0,1} \right[

$$     (Eq ) The natural and essential B.C.:
 *
 * }
 * {| style="width:100%" border="0"

$$ \displaystyle
 * style="width:95%" |
 * style="width:95%" |

\frac\left( 0 \right) = - 6

$$     (Eq )
 *
 * }
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$$ \displaystyle
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u\left( 1 \right) = 4

$$     (Eq ) Then
 *
 * }
 * {| style="width:100%" border="0"

$$ \displaystyle
 * style="width:95%" |
 * style="width:95%" |

\int {\frac{\partial }\left[ {\left( {2 + 3x} \right)\frac} \right]} dx = \int { - 5xdx}

$$     (Eq )
 *
 * }
 * {| style="width:100%" border="0"

$$ \displaystyle
 * style="width:95%" |
 * style="width:95%" |

\left( {2 + 3x} \right)\frac + {C_1} = - \frac{5}{2}{x^2}{\text{ where }}{C_1}{\text{ is a constant}}

$$     (Eq ) By the natural B.C.
 *
 * }
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$$ \displaystyle
 * style="width:95%" |
 * style="width:95%" |

{C_1} = 12

$$     (Eq ) Then
 *
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$$ \displaystyle
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\frac = \frac

$$     (Eq )
 *

 * }
 * {| style="width:100%" border="0"

$$ \displaystyle
 * style="width:95%" |
 * style="width:95%" |

u = - \frac{5}{x^2} + \frac{5}{9}x - \frac\log \left( {3x + 2} \right) + {C_2}{\text{ where }}{C_2}{\text{ is a constant}}

$$     (Eq ) By the essential B.C.
 *

 * }
 * {| style="width:100%" border="0"

$$ \displaystyle
 * style="width:95%" |
 * style="width:95%" |

{C_2} = \frac{5} - \frac{5}{9} + \frac\log \left( 5 \right) + 4

$$     (Eq ) Then
 *

 * }
 * {| style="width:100%" border="0"

$$ \displaystyle
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * style="width:92%; padding:10px; border:2px solid #8888aa" |

u = - \frac{5}{x^2} + \frac{5}{9}x - \frac\log \left( {3x + 2} \right) + \frac{5} - \frac{5}{9} + \frac\log \left( 5 \right) + 4

$$     (Eq )
 *

 * }

u=-5/12*x.^2+5/9*x+241/54*log(3*x+2)-241/54*log(2)+4;

u=-5/12*x.^2+5/9*x-118/27*log(3*x+2)+5/12-5/9+118/27*log(5)+4;

=5.6=

$$\begin{align} & b1=\frac{\left( x-{{x}_{2}} \right)\left( x-{{x}_{3}} \right)}{\left( {{x}_{1}}-{{x}_{2}} \right)\left( {{x}_{1}}-{{x}_{3}} \right)}=\frac{\left( x-2 \right)\left( x-3 \right)}{\left( 1-2 \right)\left( 1-3 \right)} \\ & b2=\frac{\left( x-{{x}_{1}} \right)\left( x-{{x}_{3}} \right)}{\left( {{x}_{2}}-{{x}_{1}} \right)\left( {{x}_{2}}-{{x}_{3}} \right)}=\frac{\left( x-1 \right)\left( x-3 \right)}{\left( 2-1 \right)\left( 2-3 \right)} \\ & b3=\frac{\left( x-{{x}_{1}} \right)\left( x-{{x}_{2}} \right)}{\left( {{x}_{3}}-{{x}_{1}} \right)\left( {{x}_{3}}-{{x}_{2}} \right)}=\frac{\left( x-1 \right)\left( x-2 \right)}{\left( 3-1 \right)\left( 3-2 \right)} \\ & \text{Assume }{{x}_{1}}=1,{{x}_{2}}=2,{{x}_{3}}=3 \\ \end{align}$$

=5.4=

title: User:Eml5526.s11.team7.jin/HW5


text: File:Phases of matter.svg

title: File:Phases of matter.svg