# A periodic ODE system

Nick Hale, November 2010

(Chebfun example ode/PeriodicSystem.m)

Chebfun can solve systems of ODEs with periodic boundary conditions. For example, consider the equation

u - v' = 0, u" + v = cos(x),

on the interval [-pi, pi] with periodic boundary conditions on u and v. A Chebfun solution could be put together like this:

d = [-pi,pi]; A = chebop(d); A.op = @(x,u,v) [u-diff(v), diff(u,2)+v]; x = chebfun('x',d); f = [0, cos(x)]; A.bc = 'periodic'; u = A\f;

We plot the result:

LW = 'linewidth'; lw = 2; FS = 'fontsize'; fs = 14; plot(u,LW,lw), title('Solutions u and v',FS,fs), legend('u','v');

For this problem, the solution can actually be computed analytically. How close were we?

true = [cos(x+3*pi/4) cos(x+pi/4)]/sqrt(2); err = norm(u-true,inf);

We show this also works for piecewise problems by artificially introducing a breakpoint at the origin.

A.domain = [-pi,0,pi]; u = A\f; plot(u,LW,lw), title('Solutions u and v',FS,fs), legend('u','v'); err = norm(u-true,inf);