Direct Quadrature for Delay Renewal Equation

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Muhammad 2024-10-23

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https://ww2.mathworks.cn/matlabcentral/answers/2161585-direct-quadrature-for-delay-renewal-equation

评论： Muhammad 2024-10-23

I'm trying to solve a delay renewal equation with a quadratic nonlinearity using direct quadrature in MATLAB. Here's the code I'm using, but I'm not getting the expected results. Can someone help me identify any mistakes or suggest improvements?

gamma = 5; 
tau = 3;   
t_min = 0; 
t_max = 20; 
dt = 0.1; 
t_vals = t_min:dt:t_max; 
phi = @(t) 0.5 * ones(size(t));  
x_vals = zeros(size(t_vals));
x_vals(1) = phi(0);  
for i = 2:length(t_vals)
    t = t_vals(i);
    integral_term = integral(@(theta) delayed(t + theta, t_vals, x_vals, phi) ...
        .* (1 - delayed(t + theta, t_vals, x_vals, phi)), -tau, -1, ...
        'RelTol', 1e-10, 'AbsTol', 1e-10);
    x_vals(i) = (gamma / 2) * integral_term;
end
figure;
plot(t_vals, x_vals, 'LineWidth', 2);
xlabel('Time');
ylabel('x(t)');
title('Solution of the Delay Renewal Equation (Direct Quadrature)');
function val = delayed(t, t_vals, x_vals, phi)
if t < 0
    val = phi(t);  
else
    val = interp1(t_vals, x_vals, t, 'linear', 'extrap');  
end
end

Thanks in advance

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Sahas 2024-10-23

Hi,

Can you share the expected results? It will help me assist you better.

Muhammad 2024-10-23

在 MATLAB Online 中打开

like here for different gamma values, different results but for my code it is not possible

gamma = 0.5;

tau = 3;

t_min = 0;

t_max = 20;

dt = 0.1;

t_vals = t_min:dt:t_max;

phi = @(t) 0.5 * ones(size(t));

x_vals = zeros(size(t_vals));

x_vals(1) = phi(0);

for i = 2:length(t_vals)

t = t_vals(i);

integral_term = integral(@(theta) delayed(t + theta, t_vals, x_vals, phi) ...

.* (1 - delayed(t + theta, t_vals, x_vals, phi)), -tau, -1, ...

'RelTol', 1e-10, 'AbsTol', 1e-10);

x_vals(i) = (gamma / 2) * integral_term;

end

Warning: Inf or NaN value encountered.

figure;

plot(t_vals, x_vals, 'LineWidth', 2);

xlabel('Time');

ylabel('x(t)');

title('Solution of the Delay Renewal Equation (Direct Quadrature)');

function val = delayed(t, t_vals, x_vals, phi)

if t < 0

val = phi(t);

else

val = interp1(t_vals, x_vals, t, 'linear', 'extrap');

end

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Direct Quadrature for Delay Renewal Equation

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