# Fredholm integral equation of the second kind with kernel containing Bessel and Struve functions

I need to solve this Fredholm integral equation of the second kind:

f[s]+integrate[f[t] K[s,t],{t,0,1}]=s


where 0<=s<=1.

The kernel is:

K[s,t]=(a/2)*(BesselJ[1,a*(s+t)]-BesselJ[1,a*Abs[s-t]]-i*StruveH[1,a*(s+t)]+i*StruveH[1,a*Abs(s-t)])


where a: real, i: imaginary unit.

I tried to solve this with the method described here: Integral equation numerical solution with NDSolve, this is the best algorithm for this case I have come across so far but it takes for ages and in the end it doesn't produce any result (due to memory insufficiency). Could anyone please help me solve it?

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There exist few typos in your kernel definition. This is how your kernel looks assuming a=2 (we denote it as A while defining the kernel as Kpart in the following).

Please utilize the code from here to solve your problem. Below I changed the constants and functional arguments of FredholmKind2 to fit your particular problem.

n = 20;(*number of discretization*)
a = 0.;
b = 1.;
lambda = -1.;
Kpart[s_, t_] := With[{A =2},
(A/2)*(BesselJ[1, A*(s + t)] - BesselJ[1, A*Abs[s - t]]-I*StruveH[1, A*(s + t)]
+ I*StruveH[1, A*Abs[s - t]])];
Gpart[x_] := x;
f1 = FredholmKind2[{a, b, lambda, Kpart, Gpart}, n,Method -> Automatic];
f2 = FredholmKind2[{a, b, lambda, Kpart, Gpart}, n,Method -> NIntegrate];


In both the methods the Re and Im part of your complex solution function coincide pretty well.

(Plot[Evaluate@(# /@ {f1[x], f2[x]}), {x, a, b}, Frame -> True,
Axes -> False, PlotStyle -> {{Thick, Opacity@.45}, {Dashed, Red}},
PlotLegends -> {"Automatic", "NIntegrate"},
ImageSize -> 400] & /@ {Re, Im})


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This was very helpful (note:the Gpart is a function of s and equal to s not x). Is there a way to solve this without evaluating A? I am asking this because next I need to calculate the function 'K[A]=A^3 Integrate[s f[s],{s,0,1}]'. – epl Aug 26 '14 at 15:18
@EvaPlevri I am afraid symbolic A will not fit here. – PlatoManiac Aug 26 '14 at 17:26
Do you know if there is another way to solve an integral equation with symbolic parameters in it? – epl Aug 27 '14 at 10:38
Sorry! I do not know answer to your symbolic parameter question. I used Plot3D[Kpart[s, t],{s,-1,1},{t,-1,1},PlotPoints-> 40] to plot the kernel. – PlatoManiac Aug 27 '14 at 11:12