Boundary condition in non-linear ODE

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Dear all,
I wanted to solve this two set of non-linear ODE using matlab :
are constant
The boundary conditions are the following :
and at
and at ( ν is an arbitrary constant < 1)
this the code that I constracted so far
function bvp4c_mathworks
rspan = [0.01 1];
init = zeros(1,4);
solinit = bvpinit(rspan,init);
sol = bvp4c(@ode4,@bc4,solinit);
eta = sol.x;
theta = sol.y(1,:);
Sr = sol.y(2,:);
plot(eta,theta)
hold on
plot(eta,Sr,'r')
hold off
legend('Nr(r)','\beta(r)')
end
function du = ode4(eta,u)
theta = u(1);
Sr = u(2); % beta
dtheta = u(3); % d(theta)/dr
dSr = u(4); % d(Sr)/dr
lambda =15.94;
P=12; %P=F*a/D; F is the applied force ; a radius of the membrane ; D = E*h^3/12(1-nu^2)
alpha = 3; %alpha =C*a^2/D ; C in-plane stifnnes
du(1) = dtheta;
du(2) = dSr;
du(3) = (P/(2*pi*eta)-(1/eta)*dtheta+(1/eta^2+lambda^2+Sr));
du(4) = (alpha*theta^2/(2*eta^2)+3/eta*dSr);
du(4) = du(4)/eta;
end
function res = bc4(u0, ur)
res = [ur(1)-0
ur(2)-0
ur(3)-0
u0(2)-0];
end
  2 Comments
Torsten
Torsten on 4 Jul 2019
du(3) and du(4) and your boundary conditions do not correspond to your mathematical equations.
zakaria azdad
zakaria azdad on 4 Jul 2019
I put theta= u(3) than du(3) = d²theta, probably I am wrong, could you give a suggestion to fix the bugg? and how could you modify the boundary conditions.
thanks

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Accepted Answer

Torsten
Torsten on 4 Jul 2019
Edited: Torsten on 4 Jul 2019
function du = ode4(eta,u)
theta = u(1);
Sr = u(2); % beta
dtheta = u(3); % d(theta)/dr
dSr = u(4); % d(Sr)/dr
lambda = 15.94;
P=12; %P=F*a/D; F is the applied force ; a radius of the membrane ; D = E*h^3/12(1-nu^2)
alpha = 3; %alpha =C*a^2/D ; C in-plane stifnnes
du = zeros(4,1);
du(1) = dtheta;
du(2) = dSr;
du(3) = P/(2*pi*eta)- 1/eta*dtheta + theta*(1/eta^2+lambda^2+Sr);
du(4) = -alpha*theta^2/(2*eta^2) - 3/eta*dSr;
end
function res = bc4(ul,ur)
nu = 0.1;
res = zeros(4,1);
res(1) = ul(1);
res(2) = ur(1);
res(3) = ul(4);
res(4) = ur(4)+(1-nu)*ur(2);
end
  9 Comments
Torsten
Torsten on 9 Jul 2019
Yes, use the solution of a converging run as initial guess for a subsequent run.
But you will have to do this on your own now because it's time to start learning MATLAB.

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