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

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ep on 25 Aug 2014
Commented: Star Strider on 27 Aug 2014
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.
The coding is from Mathematica (Matlab does not have a Struve function in its library like Mathematica does). I use this: http://www.mathworks.com/matlabcentral/fileexchange/19456-fredholm-integral-equations program to solve the integral equation but I have a problem with respect to the kernel. I haven't found how to solve symbolically the Struve function. For the Bessel one I use the Symbolic Math Toolbox and the besselj function but for the Struve one the only algorithms I have found online are ones that solve the function for a number (real,complex etc but not for a symbol). The kernel needs to be a function of s and t in order to use it in the integral. Could anyone please help me figure it out?
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### Answers (1)

Star Strider on 25 Aug 2014
I’m not familiar with Struve functions, but in my search to learn something about them, I discovered this File Exchange contribution: Struve functions. See if it does what you want.
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Star Strider on 27 Aug 2014
The only option I can consider in that instance is to keep everything symbolic and code your own Struve functions. This should not be a problem unless you have many to integrate. The Symbolic Math Toolbox is not optimal for any sort of recursion.

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