import pymatbridge
ip = get_ipython()
pymatbridge.load_ipython_extension(ip)

%%matlab
path(path,'/Users/rjl/chebfun/chebfun_v4.2.2889/chebfun')

%%matlab
x = chebfun('x');
L = chebop(@(u) diff(u,2) +  u)

%%matlab
uleft = 3.;
uright = 4;
L.lbc = uleft;
L.rbc = uright;

%%matlab
L

%%matlab
f = 0*x

%%matlab
g = L\f
plot(g,'r')
title('The Chebfun solution')

%%matlab
A = [cos(-1) sin(-1); cos(1) sin(1)];
b = [uleft; uright];
c = A\b;
u = c(1)*cos(x) + c(2)*sin(x);
error = max(abs(u-g))
plot(u)
title('The exact solution')

%%matlab
L = chebop(@(x,u) diff(u,2) - x.*u, [-30,30]);
L.lbc = 1; L.rbc = 0; u = L\0;
plot(u)
u

%%matlab
N = chebop(0,6);
N.op = @(theta) diff(theta,2) + sin(theta);


%%matlab
N.lbc = -pi/2;
N.rbc = pi/2;
theta = N\0

%%matlab
plot(-cos(theta)), grid on, ylim([-1 1])
title('Nonlinear pendulum (21.5)','fontsize',20)
xlabel('t','fontsize',20), ylabel('height -cos(\theta)','fontsize',20)

%%matlab
N.lbc = -pi/2; N.rbc = 5*pi/2; theta = N\0;
plot(-cos(theta)), grid on, ylim([-1 1])
title('Nonlinear pendulum (21.5), another solution','fontsize',20)
xlabel('t','fontsize',20), ylabel('height -cos(\theta)','fontsize',20)

%%matlab
plot(theta)

%%matlab
L = chebop(@(u) diff(u,2) + u,[0,2],@(u) u-3,@(u) diff(u)-1)

%%matlab
f = chebfun('exp(x)', [0,2]);
u = L\f

%%matlab
plot(u)

%%matlab
uprime = diff(u);
uprime(2)

%%matlab
residual = diff(uprime) + u - f;
max_residual = max(abs(residual))
plot(residual)