Question on boundary value problem

how can I deal with this boundary value problem in matlab? since I try to learn matlab by myself, I could not ask it to anyone and so I cannot genrate a sound matlab code, therefore I post nothing. Ang hint, help or suggection will be really useful to learn these problem. thank you so much for your helps!

Answer 1

[Ameer Hamza]

See the following code. The details can be quite involved, so here I just give an outline.

1. First you need to convert your 3rd order ODE to a system of 3 first order ODEs. See here for example: http://mathonline.wikidot.com/converting-nth-order-odes-to-systems-of-n-first-order-odes.

2. Then you need to write that system of ODEs into a function. See this example: https://www.mathworks.com/help/releases/R2020a/matlab/ref/bvp4c.html#mw_9083ab4d-6cde-4491-95b6-cd9b6313a459

3. Then you need to create the function to specify the boundary condition. See link in point 2 for detail.

4. Then create the initial guess. Also see link in point 2.

xmesh = linspace(0, 1, 100);
solInit = bvpinit(xmesh, [0; 0; 0]);

sol = bvp4c(@odeFun, @bcFun, solInit);
plot(sol.x, sol.y)
legend({'$y$', '$\dot{y}$', '$\ddot{y}$'}, ...
    'FontSize', 16, ...
    'Location', 'best', ...
    'Interpreter', 'latex')

function dydx = odeFun(x, y) % ODE function
    dydx = zeros(3,1);
    dydx(1) = y(2);
    dydx(2) = y(3);
    dydx(3) = -2*exp(-3*y(1)) + 4*(1+x).^-3;
end

function res = bcFun(ya, yb) % boundary Condition fcn
    res = [ya(1)-0;
           ya(2)-0;
           yb(1)-log(2)];
end

[Zeynep Toprak]

your solution is very nice! You are the best! thanks a lot...

[Ameer Hamza]

I am glad to be of help.

[Zeynep Toprak]

Dear Hamza, I have such a question, I did something, but I get wrong result, This question is similar to pde question. Can you take a look please? I like your solution, I understand perfectly :) my question is here.