create a function that solves the bvp for an arbitrary number of points without builtin functions in matlab

To create a function that solves the boundary value problem (BVP) for an arbitrary number of points without using built-in functions in MATLAB, you can use numerical methods such as the finite difference method.

Here is an example of how you can implement a function in MATLAB to solve a BVP using the finite difference method for an arbitrary number of points:

main.m
function u = solve_bvp(num_points)
% Solves a boundary value problem using the finite difference method
% for an arbitrary number of points

% Define the boundary conditions
a = 0; % start of the domain
b = 1; % end of the domain
alpha = 1; % value at the left boundary
beta = 0; % value at the right boundary

% Discretize the domain
dx = (b - a) / (num_points - 1);
x = a:dx:b;

% Create the coefficient matrix
A = zeros(num_points);
A(1, 1) = 1;
A(num_points, num_points) = 1;
for i = 2:num_points-1
    A(i, i-1) = 1/dx^2;
    A(i, i) = -2/dx^2;
    A(i, i+1) = 1/dx^2;
end

% Create the right-hand side vector
b = zeros(num_points, 1);
b(1) = alpha;
b(num_points) = beta;

% Solve the linear system of equations
u = A\b;

% Plot the solution
plot(x, u);
xlabel('x');
ylabel('u(x)');
title('Numerical Solution of the BVP');
end
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In this code snippet, the solve_bvp function takes an input num_points which specifies the number of points to use in the discretization of the domain. The function then defines the boundary conditions, discretizes the domain, creates the coefficient matrix, creates the right-hand side vector, solves the linear system of equations, and plots the solution.

You can call this function with an arbitrary number of points to solve different BVPs. For example, solve_bvp(100) will solve the BVP using 100 points.

Note that this implementation assumes a uniform grid spacing. If you want a non-uniform grid, you can modify the code accordingly. Additionally, this code assumes a second-order finite difference approximation. You can modify the stencil coefficients if you want a different approximation scheme.