Solution of second order BVP with variable functions

Hello everyone,

I am trying to solve the following 2nd order BVP:

where p and q are known functions with constant parameters. I want to solve this BVP for Y, where p and q will act as know but input functions.

Can anyone help me solve this BVP?

(For the sake of simplicity, lets consider and )

Thanks in Advance!

-Ashok Das

[Torsten]

Use "bvp4c".

Answer 1

[Sandeep Mishra]

Hi Ashok,

To solve a second-order boundary value problem (BVP) in MATLAB, you can utilize the ‘bvp4c’ and ‘bvp5c’ functions.

These functions are well-suited for handling BVPs of the form y′′(x) + sin(x)y′(x) + cos(x)y(x) = 0.

To solve the given second-order differential equation, I have made the following assumptions:

Let y1(x) = y(x) and y2(x) = y′(x), which leads to y1′(x) = y2(x) and y2′(x) = -sin(x) * y2(x) - cos(x) * y1(x).

The boundary conditions are set as: y(0) = 2 and y(1) = 3, with an initial guess for y′(0) = 0.

Refer to the following code snippet:

% Defining differential equation 
function dydx = diff_eq(x, y) 
    % dydx = [y(2); y(2)']; 
    dydx = [y(2); -sin(x) * y(2) - cos(x) * y(1)]; 
end 

% Defining boundary conditions 
function res = boundary_conditions(ya, yb) 
    % y(0) = 2, y(1) = 3 
    res = [ya(1) - 2; yb(1) - 3]; 
end 

% Initial guess for given x with y(0)=2, y’(0)=0 
solinit = bvpinit(linspace(0, 1, 10), [2 0]); 

% Solution using ‘bvp4c’ function 
sol = bvp4c(@diff_eq, @boundary_conditions, solinit); 

% Solution using ‘bvp4c’ function 
sol = bvp5c(@diff_eq, @boundary_conditions, solinit); 

For more information, refer to the following MathWorks documentation:

1. ‘bvp4c’ function: https://www.mathworks.com/help/releases/R2024a/matlab/ref/bvp4c.html
2. ‘bvp5c’ function: https://www.mathworks.com/help/releases/R2024a/matlab/ref/bvp5c.html

I hope this helps.