non-linear coupled second order ODE with matlab

Dear All,

In attempt to compare an asymptotic solution to the exact solution of Reissner theory of elasticity, I will need to solve the following coupled equation :

D, A and q are constant and the boundary conditions are of clumped boundary conditions

I would be glad if there are hints to how to implement this in matlab.

Thanks a lot!

[darova]

I tried bvp4c, but didn't get a result. Also MATLAB wants 4 boundary conditions

I did all i can, im so sorry (

[zakaria azdad]

Hi Thanks for the quick response,

The fourth boundary condition is how can I add it to the attached file?

thanks again!

[Torsten]

function res = bc4(ul, ur)
  res = [ur(1)
         ur(2)
         ur(3)
         ul(2)];
end

[zakaria azdad]

I get this error...

Error using bvp4c (line 251)

Unable to solve the collocation equations -- a singular Jacobian encountered.

[Torsten]

You divide by r^2 and r. That means that you have to start solving at r>0.

Answer 1

[darova]

According to Torsten suggestions

function bvp4c_mathworks
rspan = [0.05 1];
init = zeros(1,4);
solinit = bvpinit(rspan,init);
sol = bvp4c(@ode4,@bc4,solinit);
r = sol.x;
Nr = sol.y(1,:);
b  = sol.y(2,:);
plot(r,Nr)
hold on
plot(r,b,'r')
hold off
legend('Nr(r)','\beta(r)')
end

function du = ode4(r,u)
Nr  = u(1);
b   = u(2);             % beta
dNr = u(3);             % d(Nr)/dr
db  = u(4);             % d(beta)/dr
cb = cos(b);            % cos(beta)
sb = sin(b);            % sin(beta)
A = 0.5;
D = 1.7;
q = 1;
du(1) = dNr;
du(2) = db;
du(3) = -(3*r*dNr + Nr + (1-cb)/A)/r^2;
du(4) = -(r*Nr*sb + r^2*q*cb)/D - db + sb/r;
du(4) = du(4)/r;
end

function res = bc4(u0, ur)
res = [ur(1)-0
       ur(2)-0
       ur(3)-0
       u0(2)-0];
end

Accept my answer please

[zakaria azdad]

Hi, sorry to comeback on this point.

But how can I specify a boundary condition of this form :

as and

than I would like to evaluate

I am trying to reproduce the results of this paper (see attachement)

[darova]

You had Nr before, what is u here?

[zakaria azdad]

The authors (see attachement) define u as and

which is not clear to me who to solve it. Any hint?