Linear sine/cosine initial-value problem

Nick Trefethen and Tom Maerz, 30 September 2010

This is an elementary example to illustrate how one might use Chebfun to solve an ODE initial-value problem. We take the world's second-simplest such problem,

  u" + u = 0  ,    u(0) = 1,    u'(0) = 0

on the interval [0,100]. The solution is cos(x).

d = [0,100];                 % domain
x = chebfun('x',d);          % x variable
L = chebop(d);               % name of operator
L.op = @(u) diff(u,2) + u;   % linear operator defining the ODE
L.lbc = @(u) [u-1, diff(u)]; % imposing Dirichlet and Neumann BCs
u = L\0;                     % solve the problem
plot(u,'linewidth',1.6)      % plot the solution
err = norm(u-cos(x),inf);    % measure the error
FS = 'fontsize';
xlabel('x',FS,12)
ylabel('cos(x)',FS,12)
title(sprintf('Solution of IVP for cos(x) -- error = %7.2e',err),FS,14)
ylim([-2 2])