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## Inhomogeneous Heat Equation on Square Domain

This example shows how to solve the heat equation with a source term.

The basic heat equation with a unit source term is

` `

This is solved on a square domain with a discontinuous initial condition and zero temperatures on the boundaries.

### Create Thermal Model and Geometry

Create a transient thermal model.

```thermalmodel = createpde('thermal','transient'); ```

Create a square geometry centered at `x = 0` and `y = 0` with sides of length 2. Include a circle of radius 0.4 concentric with the square.

```R1 = [3;4;-1;1;1;-1;-1;-1;1;1]; C1 = [1;0;0;0.4]; C1 = [C1;zeros(length(R1) - length(C1),1)]; gd = [R1,C1]; sf = 'R1+C1'; ns = char('R1','C1')'; g = decsg(gd,sf,ns); ```

Append the geometry to the model.

```geometryFromEdges(thermalmodel,g); ```

### Specify Thermal Properties and Internal Heat Source

Specify thermal properties of the material.

```thermalProperties(thermalmodel,'ThermalConductivity',1,... 'MassDensity',1,... 'SpecificHeat',1); ```

Specify internal heat source.

```internalHeatSource(thermalmodel,1); ```

### Set Zero Temperatures on Boundaries

Plot the geometry and display the edge labels for use in the boundary condition definition.

```figure pdegplot(thermalmodel,'EdgeLabels','on','FaceLabels','on') axis([-1.1 1.1 -1.1 1.1]); axis equal title 'Geometry With Edge and Subdomain Labels' ``` Temperature is zero at all four outer edges of the square.

```thermalBC(thermalmodel,'Edge',1:4,'Temperature',0); ```

### Specify Initial Temperature

The discontinuous initial value is 1 inside the circle and zero outside. Specify zero initial temperature everywhere.

```thermalIC(thermalmodel,0); % Specify non-zero initial temperature inside the circle (Face 2). thermalIC(thermalmodel,1,'Face',2); ```

### Generate and Plot Mesh

```msh = generateMesh(thermalmodel); figure; pdemesh(thermalmodel); axis equal ``` ### Find and Plot Solution

Find the solution at 20 points in time between 0 and 0.1.

```nframes = 20; tlist = linspace(0,0.1,nframes); thermalmodel.SolverOptions.ReportStatistics ='on'; result = solve(thermalmodel,tlist); T = result.Temperature; ```
```99 successful steps 0 failed attempts 200 function evaluations 1 partial derivatives 20 LU decompositions 199 solutions of linear systems ```

Plot the solution.

```figure Tmax = max(max(T)); Tmin = min(min(T)); for j = 1:nframes, pdeplot(thermalmodel,'XYData',T(:,j),'ZData',T(:,j)); caxis([Tmin Tmax]); axis([-1 1 -1 1 0 1]); Mv(j) = getframe; end ``` To play the animation, use the `movie(Mv,1)` command.

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