Here, we solve a first order linear integro-differential equation considered in the Wikipedia article [1]:

$$ u'(x) + 2u(x) + 5\int_0^t u(t) dt = 1~ (x\ge 0), ~~
= 0~ (x<0) $$

with $u(0)=0$.

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 right-hand side of the integro-differential equation.

rhs = 1;

Solve the IDE using backslash.

u = N\rhs;

Here is the 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 =
8.491258814888107e-16

Plot the computed solution

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

### References

- http://en.wikipedia.org/wiki/Integro-differential_equation