User:Egm6322.s09.Three.ge/MyReport6

$$u(R,\theta,t)=0 \quad \theta \notin [\frac{-\pi}{4},\frac{\pi}{4}] \quad \forall \ t$$

Also,

$$u(r,\theta,t=0)=u_{0} \left [1-\left (\frac{r}{R} \right )^2 \right ]$$ Separation of variables applied to nonlinear PDE: Read the Handbook.

 HW: Solve the PDE:

$$f(x)(u_{x})^{2}+g(y)(u_{y})^{2}=a(x)+b(y)$$

Where:

$$\begin{matrix} f(x)=2x & g(y)=3y \\ a(x)=4x & b(y)=5y \end{matrix}$$

 HW: Summarize the handbook's general theory for separation of variables.

How to differentiate an integral.

$$\frac{d}{dx}\int_{\zeta=A(x)}^{\zeta=B(x)}F(x,\zeta)d\zeta=\int_{\zeta=A(x)}^{\zeta=B(x)}\frac{\partial }{\partial x}F(x,\zeta)d\zeta$$

Which becomes:

$$F \left[x,\zeta=B(x)\right]\frac {dB(x)}{dx}-F \left[x,\zeta=A(x)\right]\frac {dA(x)}{dx}$$

It should be noted that the principle of dimensional homogeneity must be satisfied for the previous equations.

$$\textrm{Let:} \ \textrm{  }\begin{matrix} \frac{d}{dx}\int_{\zeta=A(x)}^{\zeta=B(x)}F(x,\zeta)d\zeta=\alpha& &\int_{\zeta=A(x)}^{\zeta=B(x)}\frac{\partial }{\partial x}F(x,\zeta)d\zeta=\beta\\ F \left[x,\zeta=B(x)\right]\frac {dB(x)}{dx}=\gamma & &F \left[x,\zeta=A(x)\right]\frac {dA(x)}{dx}=\delta \end{matrix} $$

Thus,

$$[\alpha]=\textrm{the} \ \textrm{ dimension} \ \textrm{ of} \ \alpha=\frac{[F][G]}{[x]}=\beta=\gamma$$

Homogeneous and Non-homogeneous PDE's

For the general 2nd order PDE:

$$au_{xx}+2bu_{xy}+cu_{yy}+du_{x}+eu_{y}+fu+g=\mathcal D(u)$$

The differential equation is considered homogeneous if:

$$\mathcal D(u)=0$$

And non-homogeneous if:

$$\mathcal D(u)=f$$

Where f is a forcing function.

Classical Wave equation:

$$(c_{0}^{2})w_{xx}=w_{tt}$$

Exact solution, d'Alembert solution:

$$w(x,t)=\frac{1}{2} [f(x-c_{0}t)+ f(x+c_{0}t)]+\frac{1}{2c_{0}}\int_{x-c_{0}t}^{x+c_{0}t}g(\zeta)d\zeta$$

Separation of variables

The method of separation of variables seeks solve a PDE of order n by separating it into a series of n ODE's and then solving these ODE's. For a linear PDE this is done by assuming a solution that is a product of n expressions, with each expression being only a function of one independent variable.

For example, for a given PDE, the solution will be of the form:

$$u(x_{1},x_{2},x_{3},\cdots x_{n})$$

Where:

$$x_{1},x_{2},x_{3},\cdots x_{n}$$

are independent variables.

To use the method of separation of variables, one assumes that the solution may be written as a product of expressions, with each expression being a function of only one independent variable.

$$u(x_{1},x_{2},x_{3},\cdots x_{n})=X_{1}(x_{1})X_{2}(x_{2})X_{3}(x_{3}) \cdots X_{n}(x_{n})$$

Plugging this product of terms back into the PDE, one may separate the variables, and thus solve n ODE's to solve the PDE. (reference Zwillinger)

An example of Separation of Variables on a non-linear PDE is given below.