       Re: coupled integral equations

• To: mathgroup at smc.vnet.net
• Subject: [mg21206] Re: coupled integral equations
• From: Daniel Lichtblau <danl at wolfram.com>
• Date: Fri, 17 Dec 1999 01:25:03 -0500 (EST)
• Organization: Wolfram Research, Inc.
• References: <831u78\$g19@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```francesco siano wrote:
>
> Hi, I'd appreciate any help on this problem.
> I have to solve a system of integral equations of the form :
> eps1[x]=c1*eps0[x]+Integrate[kernel[x-y]*f[eps3[y]],{y,a,b}]
> eps2[x]=c2*eps0[x]+Integrate[kernel[x-y]*f[eps3[y]],{y,a,b}]
> eps3[x]=c3*eps0[x]+Integrate[kernel[x-y]*(f[eps1[y]]+f[eps2[y]]),{y,a,b}]
>
> eps4[x]=c4*eps0[x]+Integrate[kernel[x-y]*f[eps3[y]],{y,a,b}]
> (where the functions kernel, f, and the constants c1,c2,c3,c4,a,b, are
> all known) for x in the same interval [a,b].
> All I can think of is to consider [a,b] as a grid of discrete values,
> and calculate each function iteratively starting from the known
> zero-order value ci*eps0[x] (evaluating the integrals with
> ListIntegrate). This works, but very slowly. I couldn't figure out any
> smart solution. Thanks for any hint.
> -Francesco Siano
> USC

You might approximate the functions as rational functions of some fixed
degree in undetermined coefficients, then try to minimize, say, the sum
of squares of differences of left-hand-sides minus right-hand-sides of