# Time-dependent integro-differential equation

Nick Hale, October 2010

(Chebfun example pde/IntegroDiffT.m)

Here we demonstrate how to solve the time-dependent integro-differential equation

u_t = 0.02 u"(x) + \int_{-1}^{1}u(\xi)d\xi \int_{-1}^{x}u(\xi)d\xi, u(-1) = u(1) = 0.

using Chebfun's PDE15S command.

We work on the domain [-1,1] with a pulse initial condition:

```
d = [-1,1];
x = chebfun('x',d);
u0 = (1-x.^2).*exp(-30*(x+.5).^2);
```

Here is the anonymous function defining the problem for PDE15S.

f = @(u,t,x,diff,sum,cumsum) 0.02*diff(u,2) + cumsum(u)*sum(u);

The 4th, 5th, and 6th arguments define the differential, integral (sum), and indefinite integral (cumsum) operators, respectively. See 'help pde15s' for more details.

Now we solve the problem and plot the result.

[tt uu] = pde15s(f,0:.1:4,u0,'dirichlet'); FS = 'fontsize'; waterfall(uu,tt,'simple') xlabel('x',FS,14) ylabel('t',FS,14) zlabel('z',FS,14)

This example can also be found as the "Integro-differential equation" demo among the PDE-Scalar demos of CHEBGUI.