Metodos Numericos

Using the MATLAB PDE TOOLBOX
by David Hart
The MATLAB PDE Toolbox is a tool for solving two-dimensional linear partial differential equations by finite element methods. This document is intended toguide you through the use of the graphical user interface for interactive use. It assumes that the reader is running MATLAB and the PDE toolbox in a graphical environment.
At the MATLAB prompt, typepdetool. You should see a window like the one below pop up.

We will work through the solution of Laplace's equation with Dirichlet boundary conditions. Solutions of Laplace's equation, calledharmonic functions, describe situations in which forces are balanced, as might occur in electrostatics or thermal equilibria. This is an example which might be first encountered in an elementary PDEcourse, and which you might solve by Fourier series methods, by hand; you will find that you obtain the solution faster with the PDE Toolbox, and that you also obtain a visual representation, which maycontribute to the more important goal of understanding. We will then modify the example, to one which might not be so easily done by hand.
1. Draw the region of interest: suppose we want therectangle (x,y) = [0, 3] X [0, 1].
* In the Options menu, select Grid and Snap; then set the Axes' Limits to [-1 4] and [-1 2]. When finished, click Apply and Close the Axes' Limits window.

1.* In the Draw menu, select Rectangle/Square (not centered).
* In the drawing region, click and hold with the cursor at the origin, and drag it to the point with coordinates (3,1). Theregion will be shaded and marked R1.

2. Set up the boundary conditions: suppose we want the unknown solution function u to be -1 on the long horizontal sides, and +1 on the short verticalsides.
* In the Boundary menu, select Boundary Mode. The window changes to this:

*
* In the drawing region, double click on the left edge of the rectangle. The boundary condition...