Direct Quadrature for Delay Renewal Equation

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Muhammad on 23 Oct 2024

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Commented: Muhammad on 23 Oct 2024

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

2 Comments
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Sahas on 23 Oct 2024

Hi,

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

Muhammad on 23 Oct 2024

Open in MATLAB Online

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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Direct Quadrature for Delay Renewal Equation

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