3
$\begingroup$

I have integro-differential equations like this:

γ = 0.1;
κ = 0.15;
g = 0.2;
δ = 0.2 + 0.6 I;

eqns = {
   x'[t] == -γ x[t] - g Re@z[t],
   y'[t] == -κ y[t] + g Re@z[t],
   z[t] == 
    Integrate[(x[τ] - 
        y[τ]) Exp[ -δ (t - τ)], {τ, 0, t}]
   };

ints = {
   x[0] == 1,
   y[0] == 0
   };

NDSolve[Join[eqns, ints], {x, y}, {t, 0, 10}]

I don't know how to use Mathematica to solve it or if it can be solved at all using some combinations of built-in functions?

To solve integro-differential equations in Mathematica is important to me for studying some special physical models.

$\endgroup$
  • $\begingroup$ Related: mathematica.stackexchange.com/questions/24626/… $\endgroup$ – Michael E2 Oct 10 '13 at 13:00
  • $\begingroup$ Solving integral equations is hard enough. In general there is no systematic approach. Look e.g. here: How to solve system of integral equations how one can get a general idea of possible solutions. There is no built-in functionality in any computer system for solving inegro-differential equations as far as I can say. $\endgroup$ – Artes Oct 10 '13 at 13:20
  • $\begingroup$ Also might want to check responses to similar question here $\endgroup$ – Daniel Lichtblau Oct 10 '13 at 18:28
1
$\begingroup$

For your special problem, it seems you can differentiate the third equation and transform it into the differential equation x[t] - y[t] - δ z[t]==z'[t]. You can also deduce boundary conditions from the integral equation (compute z[0]).

$\endgroup$
  • $\begingroup$ Hi Ahmed, welcome to Mathematica StackExchange! Don't forget to upvote good answers (and other people's questions) using the triangle above the number next to the post. I edited the formatting of your answer as per the guide found on the help centre. It's a good idea to read the guide and the about page $\endgroup$ – gpap Oct 10 '13 at 20:41

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.