0.6 Solution of the partial differential equations  (Page 3/13)

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$\begin{array}{c}\frac{\partial \mathbf{v}}{\partial t}=\mathbf{f}-\frac{\nabla p}{\rho }+\nu \phantom{\rule{0.166667em}{0ex}}{\nabla }^{2}\mathbf{v},\phantom{\rule{1.em}{0ex}}\mathrm{velocity}\phantom{\rule{0.277778em}{0ex}}\mathrm{perpendicular}\phantom{\rule{0.277778em}{0ex}}\mathrm{to}\phantom{\rule{0.277778em}{0ex}}\mathrm{velocity}\phantom{\rule{0.277778em}{0ex}}\mathrm{gradient}\hfill \\ \\ \frac{\partial \mathbf{v}}{\partial t}=\nu \phantom{\rule{0.166667em}{0ex}}{\nabla }^{2}\mathbf{v},\phantom{\rule{1.em}{0ex}}\mathrm{if}\phantom{\rule{0.277778em}{0ex}}\mathbf{f}\phantom{\rule{0.277778em}{0ex}}\mathrm{and}\phantom{\rule{0.277778em}{0ex}}\nabla p\phantom{\rule{0.277778em}{0ex}}\mathrm{vanish}\hfill \end{array}$

Green's function, convolution, and superposition

A property of linear PDEs is that if two functions are each a solution to a PDE, then the sum of the two functions is also a solution of the PDE. This property of superposition can be used to derive solutions for general boundary, initial conditions, or distribution of sources by the process of convolution with a Green's function. The student is encouraged to read P. M. Morse and H. Feshbach, Methods of Theoretical Physics , 1953 for a discussion of Green's functions.

The Green's function is used to find the solution of an inhomogeneous differential equation and/or boundary conditions from the solution of the differential equation that is homogeneous everywhere except at one point in the space of the independent variables. (The initial condition is considered as a subset of boundary conditions here.) When the point is on the boundary, the Green's function may be used to satisfy inhomogeneous boundary conditions; when it is out in space, it may be used to satisfy the inhomogeneous PDE.

The concept of Green's solution is most easily illustrated for the solution to the Poisson equation for a distributed source $\rho \left(x,y,z\right)$ throughout the volume. The Green's function is a solution to the homogeneous equation or the Laplace equation except at $\left({x}_{o},{y}_{o},{z}_{o}\right)$ where it is equal to the Dirac delta function. The Dirac delta function is zero everywhere except in the neighborhood of zero. It has the following property.

${\int }_{-\infty }^{\infty }f\left(\xi \right)\phantom{\rule{0.166667em}{0ex}}\delta \left(\xi -x\right)\phantom{\rule{0.166667em}{0ex}}d\xi =f\left(x\right)$

The Green's function for the Poisson equation in three dimensions is the solution of the following differential equation

$\begin{array}{c}{\nabla }^{2}G=-\delta \left(\mathbf{x}-{\mathbf{x}}_{o}\right)=-\phantom{\rule{0.166667em}{0ex}}\delta \left(x-{x}_{o}\right)\phantom{\rule{0.166667em}{0ex}}\delta \left(y-{y}_{o}\right)\phantom{\rule{0.166667em}{0ex}}\delta \left(z-{z}_{o}\right)\hfill \\ G\left(\mathbf{x}\left|{\mathbf{x}}_{o}\right)\right)=\frac{1}{4\pi \left|\mathbf{x}-{\mathbf{x}}_{o}\right|}\hfill \end{array}$

It is a solution of the Laplace equation except at $\mathbf{x}={\mathbf{x}}_{o}$ where it has a singularity, i.e., it has a point source. The solution of the Poisson equation is determined by convolution.

$u\left(\mathbf{x}\right)=\phantom{\rule{0.277778em}{0ex}}\int \int \int \phantom{\rule{0.277778em}{0ex}}G\left(\mathbf{x}\left|{\mathbf{x}}_{o}\right)\right)\phantom{\rule{0.166667em}{0ex}}\rho \left({\mathbf{x}}_{o}\right)\phantom{\rule{0.166667em}{0ex}}d{x}_{o}\phantom{\rule{0.166667em}{0ex}}d{y}_{o}\phantom{\rule{0.166667em}{0ex}}d{z}_{o}$

Suppose now that one has an elliptic problem in only two dimensions. One can either solve for the Green's function in two dimensions or just recognize that the Dirac delta function in two dimensions is just the convolution of the three-dimensional Dirac delta function with unity.

$\delta \left(x-{x}_{o}\right)\phantom{\rule{0.166667em}{0ex}}\delta \left(y-{y}_{o}\right)={\int }_{-\infty }^{\infty }\delta \left(x-{x}_{o}\right)\phantom{\rule{0.166667em}{0ex}}\delta \left(y-{y}_{o}\right)\phantom{\rule{0.166667em}{0ex}}\delta \left(z-{z}_{o}\right)\phantom{\rule{0.166667em}{0ex}}d{z}_{o}$

Thus the two-dimensional Green's function can be found by convolution of the three dimensional Green's function with unity.

$\begin{array}{ccc}\hfill G\left(x,y|{x}_{o},{y}_{o}\right)& =& {\int }_{-\infty }^{\infty }G\left(\mathbf{x}|{\mathbf{x}}_{o}\right)\text{\hspace{0.17em}}d{z}_{o}\hfill \\ & =& -\frac{1}{4\pi }\mathrm{ln}\left[{\left(x-{x}_{o}\right)}^{2}+{\left(y-{y}_{o}\right)}^{2}\right]\hfill \end{array}$

This is a solution of the Laplace equation everywhere except at $\left({x}_{o},{y}_{o}\right)$ where there is a line source of unit strength. The solution of the Poisson equation in two dimensions can be determined by convolution.

$u\left(x,y\right)=\iint G\left(x,y|{x}_{o},{y}_{o}\right)\text{\hspace{0.17em}}\rho \left({x}_{o},{y}_{o}\right)\text{\hspace{0.17em}}d{x}_{o}\text{\hspace{0.17em}}d{y}_{o}$

Assignment 7.2 derivation of the green's function

Derive the Green's function for the Poisson equation in 1-D, 2-D, and 3-D by transforming the coordinate system to cylindrical polar or spherical polar coordinate system for the 2-D and 3-D cases, respectively. Compare the results derived by convolution.

Method of images

Green's functions can also be determined for inhomogeneous boundary conditions (the boundary element method) but will not be discussed here. The Green's functions discussed above have an infinite domain. Homogeneous boundary conditions of the Dirichlet type $\left(u=0\right)$ or Neumann type $\left(\partial u/\partial n=0\right)$ along a plane(s) can be determined by the method of images.

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