solve 3rd order differential equation

Dear Matlab Users,

I am struggling for solving a complicated 3rd order differential equation. Kindly suggest some guidelines for solving following differential equation with inintial conditions, h=0.1, h' = 0.3, h'' = 1. I refer ode45, solvng differential equation pages, here I am not able to convert this equation in the form of h'''+h''+h'+h = f(r).

Thank you.

Yashika

Answer 1

[Ameer Hamza]

If you have the symbolic toolbox, then you can use odeToVectorField to convert your ODE into the standard form of H' = f(r, H) needed for ode45. For example

syms h(r)

term1 = (diff(h, r, 2) + 1/r*diff(h, r, 1) + 1/r*diff(h, r, 1)^3)/(1 + diff(h, r, 1)^2)^(3/2);
rhs = 2*r - r^2*h^3*diff(term1, r, 1);
lhs = h/r + diff(h, r, 1);

eq = lhs == rhs;

V = odeToVectorField(eq);
odefun = matlabFunction(V, 'Vars', {'r', 'Y'}); % Y denotes h here.

IC = [0.1; 0.3; 1];
rspan = [0 10];
[r_sol, h_sol] = ode45(odefun, rspan, IC);

However, there seems to be a singularity in the ODE, or there is a typing mistake, the ode45 gives NaN output.

[Yashika]

Dear Ameer,

I am extremely grateful for your humble reply and for the code with explaination. I checked it, equation has no typing mistake. There could be sigularity at some point. I will check for the r domain where non-singular solution occur.

Regards,

Yashika

[Ameer Hamza]

I am glad to be of help!