# A Solving two simultaneous integro-differential equations

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1. Oct 13, 2016

### Boudy

I am trying to find a closed-form (analytical) solution for the two following simultaneous integro-differential equations :

du[x]/dx= - a v[x] +b ∫〖[1-(y-x)^4 〗].(v[y]-v[x])dy
And
(dv[x])/dx= - f u[x] -g ∫〖[1-(y-x)^4 〗].u[y]dy
With the initial conditions:
v[0]=e and u[1]=0
a,b,f,g and e are positive constants
Both integrals are from y=0 to y=1.
The unknowns are u[x] and v[x].

2. Oct 13, 2016

### Dr Transport

an analytical expression for that set of equations is most likely not going to be found.

3. Oct 21, 2016

### ModestyKing

Just wrote down the start to a solution.
Differentiate one of the equations with respect to x. You already have expressions for du/dx and dv/dx, so when you differentiate you can substitute the other equation in. Voila! Suddenly you've got an integro-differential equation for only one function. I'll leave you to do the rest of it ;)

4. Oct 27, 2016

### Boudy

I thank both friends for their kind help. Fortunately, and after some modifications in the equations, a solution was possible using the numerical solution of two simultaneous differential equations in two variables and one single independent variable. This was done using the NDSolve command within a Mathematica 11 code.
Thanks again.