Wikipedia integro-differential equation example

Mark Richardson, 27 September 2010

Here, we solve a first order linear integro-differential equation considered in the Wikipedia article:

      u'(x) + 2u(x) + 5\int_{0}^{t} u(t) dt = 1,     x >= 0
                                            = 0,     x  < 0
                  u(0) = 0

Begin by defining the domain d, chebfun variable x and operator N.

d = [0 5];
x = chebfun('x',d);
N = chebop(d);

The problem has a single Dirichlet boundary condition at x = 0.

N.lbc = 0;

Define the operator using Chebfun's overloaded DIFF and CUMSUM commands.

N.op = @(u) diff(u) + 2*u + 5*cumsum(u);

Set the RHS of the IDE.

rhs = 1;

Solve the IDE using backslash.

u = N\rhs;

Analytic solution:

u_exact = 0.5*exp(-x).*sin(2*x);

How close is the computed solution to the true solution?

accuracy = norm(u-u_exact)

accuracy =
     3.282457072036605e-16

Plot the computed solution

plot(u,'linewidth',1.6), grid on
title('Solution of integro-differential equation','fontsize',16)