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Vida Afosaa Sure, I can help you generate the Haar wavelet operational matrix of integration of order one in MATLAB. To do this, we need to follow these steps:

Here's the MATLAB code to achieve this:

function P = haar_wavelet_operational_matrix(order, J)

% Check if the order is 1 (integration)

if order ~= 1

error('Order other than 1 is not supported for Haar wavelet operational matrix.');

end

% Calculate the number of intervals, 2^J

M = 2^J;

% Initialize the operational matrix

P = zeros(2*M);

% Generate Haar wavelet function of order one

x = linspace(0, 1, 2*M+1);

phi = zeros(1, 2*M+1);

for i = 1:length(x)

if x(i) >= 0 && x(i) < 0.5

phi(i) = 1;

elseif x(i) >= 0.5 && x(i) < 1

phi(i) = -1;

end

end

% Compute the values of the Haar wavelet function at certain points

% within the domain [0, 1].

for i = 1:2*M

% Integrate phi(x) from 0 to x and store the result in the matrix

P(i, 1) = integral(@(t) haar_wavelet(t, phi, x), 0, x(i));

% Integrate phi(x) from x to 1 and store the result in the matrix

P(i, 2*M) = integral(@(t) haar_wavelet(t, phi, x), x(i), 1);

end

end

function y = haar_wavelet(t, phi, x)

% This function computes the value of Haar wavelet function at t

% using the given values phi and x.

y = 0;

for j = 1:length(x)-1

if t >= x(j) && t < x(j+1)

y = phi(j);

break;

end

end

end

This code defines two functions: haar_wavelet_operational_matrix and haar_wavelet. The first function generates the Haar wavelet operational matrix of integration of order one for a given maximum level of resolution J. The second function computes the Haar wavelet function at a given point t using the given values phi and x.

You can call the haar_wavelet_operational_matrix function with the desired order (1) and maximum level of resolution (J) to get the Haar wavelet operational matrix of integration of order one. For example:

J = 3; % Maximum level of resolution

P = haar_wavelet_operational_matrix(1, J);

disp(P);

The resulting matrix P will be a 2M x 2M square matrix, where M is 2^J, as mentioned in your question.