% Define the ODE function according to your problem statement.
% It's important to ensure that this function correctly models the physical or mathematical system.
fun = @(x, y, mu, lambda, ke, ko) [x * y(2);
(-x^2 * y(2) + x * y(1) - (ke^2 * x^4) / (2 * mu + lambda) + (ko^2 * x^4 * y(3)) / (2 * mu + lambda)) / x^2;
x * y(4);
(-x^2 * y(4) + x * y(3) - (ko^2 * x^4) / mu * y(1) - (ke^2 * x^4) / mu * y(3)) / x^2];
% Define your initial conditions and ensure they are suitable for the problem.
initialConditions = [0.2; 0; 0; 0];
% Define parameter ranges. Ensure these ranges are logical for your problem.
ke_range = linspace(0.0001, 0.6, 10);
ko_range = linspace(0.0001, 0.6, 10);
% Preallocate matrices to store results for efficiency.
peak_r = zeros(length(ke_range), length(ko_range));
peak_theta = zeros(length(ke_range), length(ko_range));
% Loop through each combination of parameters.
for m = 1:length(ke_range)
for n = 1:length(ko_range)
ke = ke_range(m);
ko = ko_range(n);
% Define material properties or other parameters if they vary.
mu = 10; % Example value
lambda = 50; % Example value
% Solve the differential equations over a defined x domain.
% Adjust the x domain [0.75, 210] as necessary for your problem.
[x, y] = ode45(@(x, y) fun(x, y, mu, lambda, ke, ko), [0.75, 210], initialConditions);
% Extract and store the peak values for r(x) and theta(x).
% This method captures the peaks in the results for each ke, ko pair.
peak_r(m, n) = max(abs(y(:, 1)));
peak_theta(m, n) = max(abs(y(:, 3)));
end
end
% Visualization
% Create 3D surface plots to visualize how peak values change with ke and ko.
figure;
surf(ke_range, ko_range, peak_r');
title('Peak Extremum Values of r(x)');
xlabel('ke');
ylabel('ko');
zlabel('Peak r(x)');
colorbar;
figure;
surf(ke_range, ko_range, peak_theta');
title('Peak Extremum Values of Theta(x)');
xlabel('ke');
ylabel('ko');
zlabel('Peak Theta(x)');
colorbar;
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It's important to note that the advice and code are based on limited information and meant for educational purposes. Users should verify and adapt the code to their specific needs, ensuring compatibility and adherence to ethical standards.
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