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How to solve a class of integro-differential equations system?

Hi everyone

I'm trying to solve a particular physical problem that leads to a system of integro-differential equation that never I found neither solved. The system is described by two state variables z(t) and $\Phi_2(t)$ ; $\Phi_1$ , $\rho$, $\pi$, Mr, nF and R are known constants. In the convolution integral, the functions v(t-$\tau$) and f(z) are known; the last one is coupled with the solution of the first differential equation. The function j( $\tau$) is also a solution of other differential equation or can be written as linear combination of known functions.

I would like use the native functions of Mathematica to solve this system before to try an Euler or collocation method. If someone has experience in this kind of problems I would appreciate any suggestion and comments.

Regards Javier Navarro

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I think your problem is the initial condition z(0) = 0 which make dz/dt(0) blow up for any nonzero Phi1 and Phi2(0)=0

POSTED BY: Kay Herbert

Hello Kay. Thanks for your comment. No, the initial condition z(0)=0 is not a problem because initially the first derivative tends to infinity; that's compatible with the physical problem. The way I by-passed z(0)=0 is to consider z(0) ~ $10^{-15}$. Regards Javier

Hello Daniel, Thanks for your comments. Sometimes you have the answer in front of you and you don't realize. Yes the derivative works for some particular situations. Thanks, Regards Javier

P.S. : I upload the original Mathematica file

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(1) Why not differentiate the second equation to get rid of the integral operator?

(2) Always better if the equations provided are in Mathematica input form. Then people can cut and paste them for purposes of attempting to answer questions.

POSTED BY: Daniel Lichtblau
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