Solve the higher order of boundary condition.

How to incorporate higher-order boundary conditions into the matrix of the Keller box method?

[Torsten]

Since you always have to reduce higher-order equations/systems to a first-order system before starting to solve it, "higher-order boundary conditions" don't exist. Or what exactly do you mean ?

Answer 1

[SAI SRUJAN]

Hi Nur,

I understand that you are trying to handle higher-order boundary conditions when using the Keller box method to solve differential equations.

Convert higher-order ordinary differential equations (ODEs) into a system of first-order ODEs, once we have the system of first-order ODEs and the corresponding boundary conditions, we can use the following MATLAB functions to solve the systems of ODEs. Here are some MATLAB functions that might be used:

• 'bvp4c' or 'bvp5c': These functions are designed for solving boundary value problems (BVPs) for systems of ODEs. They are well-suited for problems with two-point boundary conditions.
• 'fsolve': This function can be used to solve systems of nonlinear algebraic equations, which is what you get after discretizing a system of ODEs with the Keller box method.

Please follow the below example to understand the working of 'bvp4c',

function example
    function dydx = diff_eqn(x, y)
        dydx = [y(2); -y(1)]; % Example: y'' + y = 0
    end

    function res = bc(ya, yb)
        res = [ya(1) - alpha; yb(1) - beta]; % Example: y(0) = alpha, y(1) = beta
    end

    solinit = bvpinit(linspace(0, 1, 10), [0, 0]);

    sol = bvp4c(@diff_eqn, @bc, solinit);
end

For a comprehensive understanding of the 'fsolve' and 'bvp4c' functions in MATLAB, please refer to the following documentation.

https://www.mathworks.com/help/optim/ug/fsolve.html

https://www.mathworks.com/help/matlab/ref/bvp4c.html

I hope this helps!