how to solve a system of integro-differential equations?

Answer 1

[Hewa selman]

[Torsten]

Since

integral_{0}^{t} exp(t-s) ds = exp(t) - 1,

you can use any of the ODE solvers MATLAB supplies (e.g. ODE45, ODE15S).

I did not yet test the code, but if your problem is more complicated, maybe the link Claudio Gelmi gave under

https://de.mathworks.com/matlabcentral/answers/225262-solving-integro-differential-equation-with-limited-integral

is useful for you.

[Hewa selman]

Thank you.

So, directly there is no code for solving system of integro differential equations .

[Hewa selman]

Yes, it is more complicated and I need a code for solve it.

I found the code idsolver, for solving integro differential equations, but I am not sure that it used for the system of integro differential equations too.

Answer 2

[John D'Errico]

Last time I checked, int(exp(-t),[-inf,T]) is both trivial to integrate, and yet essentially infinite, for ANY finite value of T. So your system of equation is poorly defined. Do I really need to show this?

syms t T
int(exp(-t),[-inf,T])

ans = 

∞

That says your system of equations is not solvable.

[Hewa selman]

Thank you for your notation.

So if I change it a bit to the following form, can I find a code in Matlab to solve the system of the integro-differential equations?

\frac{dx}{dt}=tx(t)+y(t)+sin(y)\int_{0}^t e^{(t-s)} ds \\

x_{0}=0\\

\frac{dy}{dt}=x(t)+ty(t)+cos(x)\int_{0}^t e^{(t-s)} ds \\

y_{0}=1

With regards