Solving third order nonlinear boundary value problem

I have a boundary value problem in this form:

where alpha is a nonzero constant and boudary conditions are:

f(-1) = 0

f(0) = 1

f(1) = 0

How can I solve this problem using bvp5c?

[David Goodmanson]

Hi parham,

This looks doable if you constuct an f vector of length 6, where the first three elements are f,f',f'' in the left hand region -1<=x<=0 and the second three elements are f,f',f'' in the right hand region 0<=x<=1. Then the differential equation part will be

dfdt(1) = f(2)  
dfdt(2) = f(3)  
dfdt(3) = a*f(1)*f(2) + 4*a^2*f(2)  

and the same for f variables (4,5,6) since the differential equation is the same. At the boundary x = 0, the first and second derivatives of f have to be continuous across the boundary. So for the boundary part in terms of fa and fb, remembering that the a,b boundaries are -1,0 for f (1,2,3) and 0,1 for f(4,5,6) ,the follwing values are zero:

fa(1)
fb(1)-1
fa(4)-1
fb(4)
fb(2)-fa(5)  % continuous 1st derivative
fb(3)-fa(6)  % continuous 2nd derivative

[PRITESH]

Hi @David Goodmanson

I have the exact same equation, and I understood everything about your code, except for one thing.

Instead of dfdt(3) = a*f(1)*f(2) + 4*a^2*f(2), shouldn't it be dfdt(3) = -a*f(1)*f(2) - 4*a^2*f(2) ?

[Torsten]

Instead of dfdt(3) = a*f(1)*f(2) + 4*a^2*f(2), shouldn't it be dfdt(3) = -a*f(1)*f(2) - 4*a^2*f(2) ?

Yes.

[PRITESH]

Hi @Torsten. Thanks for the reply. I am still confused while applying this technique. I am trying to use bvp4c and am getting singular Jacobian values.

[Torsten]

I am trying to use bvp4c and am getting singular Jacobian values.

I don't:

alpha = 1;
xc = 0;
xmesh = [linspace(-1,xc,1000),linspace(xc,1,1000)];
solinit = bvpinit(xmesh, [0 0 0]);
sol = bvp5c(@(x,y,r)f(x,y,r,alpha),@bc,solinit);
plot(sol.x,sol.y(1,:))

function dydx = f(x,y,region,alpha)
  dydx = [y(2);y(3);-alpha*y(1)*y(2)-4*alpha^2*y(2)];
end

function res = bc(yl,yr)
   res = [yl(1,1);yr(1,1)-1;yr(1,1)-yl(1,2);yr(2,1)-yl(2,2);yr(3,1)-yl(3,2);yr(1,2)];
end