# Documentation

## Enter Coefficients in the PDE App

This example shows how to enter coefficients in the PDE app.

Caution: Do not include spaces in your coefficient strings in the PDE app. The string parser can misinterpret a space as a vector separator, as when a MATLAB® vector uses a space to separate elements of a vector.

The PDE is parabolic,

$d\frac{\partial u}{\partial t}-\nabla \cdot \left(c\nabla u\right)+au=f,$

with the following coefficients:

• d = 5

• a = 0

• f is a linear ramp up to 10, holds at 10, then ramps back down to 0:

$f=10*\left\{\begin{array}{ll}10t\hfill & 0\le t\le 0.1\hfill \\ 1\hfill & 0.1\le t\le 0.9\hfill \\ 10-10t\hfill & 0.9\le t\le 1\hfill \end{array}$

• c = 1 +.x2 + y2

These coefficients are the same as in Solve PDE with Coefficients in Functional Form.

Write the following file `framp.m` and save it on your MATLAB path.

```function f = framp(t) if t <= 0.1 f = 10*t; elseif t <= 0.9 f = 1; else f = 10-10*t; end f = 10*f;```

Open the PDE app, either by typing `pdetool` at the command line, or selecting PDE from the Apps menu.

Select PDE > PDE Specification.

Select Parabolic equation. Fill in the coefficients as pictured:

• c = `1+x.^2+y.^2`

• a = `0`

• f = `framp(t)`

• d = `5`

The PDE app interprets all inputs as strings. Therefore, do not include quotes for the `c` or `f` coefficients.

Select Options > Grid and Options > Snap.

Select Draw > Draw Mode, then draw a rectangle centered at (0,0) extending to 1 in the x-direction and 0.4 in the y-direction.

Draw a circle centered at (0.5,0) with radius 0.2

Change the set formula to `R1-C1`.

Select Boundary > Boundary Mode

Click a segment of the outer rectangle, then Shift-click the other three segments so that all four segments of the rectangle are selected.

Double-click one of the selected segments.

Fill in the resulting dialog box as pictured, with Dirichlet boundary conditions h = `1` and r = `t*(x-y)`. Click OK.

Select the four segments of the inner circle using Shift-click, and double-click one of the segments.

Select Neumann boundary conditions, and set g = `x.^2+y.^2` and q = 1. Click OK.

Click to initialize the mesh.

Click to refine the mesh. Click again to get an even finer mesh.

Select Mesh > Jiggle Mesh to improve the quality of the mesh.

Set the time interval and initial condition by selecting Solve > Parameters and setting Time = `linspace(0,1,50)` and u(t0) = `0`. Click OK.

Solve and plot the equation by clicking the button.

Match the following figure using Plot > Parameters.

Click the Plot button.