Direct Quadrature for Delay Renewal Equation

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Muhammad
Muhammad on 23 Oct 2024
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
Sahas
Sahas on 23 Oct 2024
Hi,
Can you share the expected results? It will help me assist you better.
Muhammad
Muhammad on 23 Oct 2024
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.
Warning: Inf or NaN value encountered.
Warning: Inf or NaN value encountered.
Warning: Inf or NaN value encountered.
Warning: Inf or NaN value encountered.
Warning: Inf or NaN value encountered.
Warning: Inf or NaN value encountered.
Warning: Inf or NaN value encountered.
Warning: Inf or NaN value encountered.
Warning: Inf or NaN value encountered.
Warning: Inf or NaN value encountered.
Warning: Inf or NaN value encountered.
Warning: Inf or NaN value encountered.
Warning: Inf or NaN value encountered.
Warning: Inf or NaN value encountered.
Warning: Inf or NaN value encountered.
Warning: Inf or NaN value encountered.
Warning: Inf or NaN value encountered.
Warning: Inf or NaN value encountered.
Warning: Inf or NaN value encountered.
Warning: Inf or NaN value encountered.
Warning: Inf or NaN value encountered.
Warning: Inf or NaN value encountered.
Warning: Inf or NaN value encountered.
Warning: Inf or NaN value encountered.
Warning: Inf or NaN value encountered.
Warning: Inf or NaN value encountered.
Warning: Inf or NaN value encountered.
Warning: Inf or NaN value encountered.
Warning: Inf or NaN value encountered.
Warning: Inf or NaN value encountered.
Warning: Inf or NaN value encountered.
Warning: Inf or NaN value encountered.
Warning: Inf or NaN value encountered.
Warning: Inf or NaN value encountered.
Warning: Inf or NaN value encountered.
Warning: Inf or NaN value encountered.
Warning: Inf or NaN value encountered.
Warning: Inf or NaN value encountered.
Warning: Inf or NaN value encountered.
Warning: Inf or NaN value encountered.
Warning: Inf or NaN value encountered.
Warning: Inf or NaN value encountered.
Warning: Inf or NaN value encountered.
Warning: Inf or NaN value encountered.
Warning: Inf or NaN value encountered.
Warning: Inf or NaN value encountered.
Warning: Inf or NaN value encountered.
Warning: Inf or NaN value encountered.
Warning: Inf or NaN value encountered.
Warning: Inf or NaN value encountered.
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
end

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