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

Caution: Do not include spaces when you specify your coefficients
the PDE app. The 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:$$f=10*\{\begin{array}{ll}10t\hfill & 0\le t\le 0.1\hfill \\ 1\hfill & 0.1\le t\le 0.9\hfill \\ 10-10t\hfill & 0.9\le t\le 1\hfill \end{array}$$

*c*= 1 +.*x*^{2}+*y*^{2}

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 vectors of characters. 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.

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