function phi = haar_wavelet(x, j, k)

% Haar wavelet function

% j: scale

% k: translation

% x: point where the function is evaluated

phi = (x >= k/2^(j) && x < (k+1)/2^(j)) - (x >= (k+1)/2^(j) && x < (k+2)/2^(j));

end

function [A, b] = setup_collocation_matrix(N, L, j_max)

% N: number of collocation points

% L: length of the domain

% j_max: maximum scale of the Haar wavelets

% Define the collocation points

x = linspace(0, L, N);

% Initialize the matrix and right-hand side vector

A = zeros(N, 2^(j_max));

b = zeros(N, 1);

% Construct the matrix A and vector b

for i = 1:N

for j = 0:j_max-1

for k = 0:2^j-1

A(i, j*2^j + k + 1) = haar_wavelet(x(i), j, k);

end

end

end

% Define the right-hand side vector b based on the PDE

% Example: For a simple PDE u_xx = -2 * pi^2 * sin(pi*x)

b = -2 * pi^2 * sin(pi * x);

end

function u = haar_wavelet_solver(N, L, j_max)

% Set up the collocation matrix and right-hand side vector

[A, b] = setup_collocation_matrix(N, L, j_max);

% Solve the system of linear equations

coeff = A \ b;

% Reconstruct the solution

x = linspace(0, L, N);

u = zeros(size(x));

for i = 1:N

for j = 0:j_max-1

for k = 0:2^j-1

u(i) = u(i) + coeff(j*2^j + k + 1) * haar_wavelet(x(i), j, k);

end

end

end

end

% Parameters

N = 100; % Number of collocation points

L = 1; % Length of the domain

j_max = 4; % Maximum scale of Haar wavelets

% Solve the PDE

u = haar_wavelet_solver(N, L, j_max);

% Plot the solution

x = linspace(0, L, N);

figure;

plot(x, u, '-o');

title('Solution using Haar Wavelet Collocation Method');

xlabel('x');

ylabel('u(x)');

grid on;