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

Question

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?

Answer 1

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})