System of two nonlinear BVPs
Asgeir Birkisson, September 2010
Contents
(Chebfun example ode/BVPSystem.m)
1. System of equations
Here is a system of two coupled nonlinear ODEs on the interval [-1,1], with boundary conditions.
u'' - sin(v) = 0
v'' + cos(u) = 0
u(-1) = 1, v'(-1) = 0
u'(1) = 0, v(1) = 0
2. Solution using multiple variables u and v
One way you can solve a problem like this with Chebfun is to work with multiple variables, solving for two chebfuns u and v. Here we do this, setting up the problem using anonymous functions that take two chebfuns as input and return a quasimatrix of two chebfuns as output:
N = chebop(-1,1);
x = chebfun('x');
N.op = @(x,u,v)[ diff(u,2) - sin(v), diff(v,2) + cos(u)];
N.lbc = @(u,v)[ u-1, diff(v)];
N.rbc = @(u,v)[ v, diff(u)];
N.guess = [0*x,0*x];
[sol nrmduvec] = N\0;
We extract the functions from the solution and plot them:
LW = 'linewidth'; FS = 'fontsize'; u = sol(:,1); v = sol(:,2); figure, subplot(1,2,1), plot(u,LW,2) hold on, plot(v,'--r',LW,2), hold off title('u and v vs. x',FS,12), legend('u','v') box on, grid on xlabel('x',FS,10), ylabel('u(x) and v(x)',FS,10) subplot(1,2,2), semilogy(nrmduvec,'-*',LW,2) title('Norm of update vs. iteration no.',FS,12) box on, grid on xlabel('Iteration no.',FS,10), ylabel('Norm of update',FS,10)
3. Solution using a single quasimatrix variable u
Another way to solve the same problem is to work with a single quasimatrix variable u which has two components, u(:,1) and u(:,2).
(u_1)'' - sin(u_2) = 0
(u_2)'' + cos(u_1) = 0
u_1(-1) = 1, (u_2)'(-1) = 0
(u_1)'(1) = 0, u_2(1) = 0
N = chebop(-1,1); x = chebfun('x'); N.op = @(u) [ diff(u(:,1),2) - sin(u(:,2)), diff(u(:,2),2) + cos(u(:,1)) ]; N.lbc = @(u)[ u(:,1)-1, diff(u(:,2))]; N.rbc = @(u)[ u(:,2), diff(u(:,1))]; N.guess = [0*x,0*x]; [u nrmduvec] = N\0; figure subplot(1,2,1), plot(u(:,1),LW,2), hold on plot(u(:,2),'--r',LW,2), hold off title('u_1(x) and u_2(x) vs. x',FS,12), legend('u_1','u_2') box on, grid on xlabel('x',FS,10), ylabel('u_1(x) and u_2(x)',FS,10) subplot(1,2,2), semilogy(nrmduvec,'-*',LW,2) title('Norm of update vs. iteration no.',FS,12) box on, grid on xlabel('Iteration no.',FS,10), ylabel('Norm of update',FS,10)
