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.

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?

Answer 1

[Star Strider]

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.

[ep]

Thank you for your answer but this program requires the argument to be a complex vector/matrix, it doesn't accept symbols.

[Star Strider]

If you are integrating the function numerically, as a function handle (from a function file or anonymous function), you can integrate it with the integral function (or its friends, integral2 or integral3), with respect to both variables and without the use of symbols.

If you have already evaluated the function (perhaps using meshgrid first), then you can use trapz to integrate it on as many dimensions (sequentially) as you need to.

What is your MATLAB code for your calculations?

[ep]

The problem lies in the fact that "a" in the kernel needs to remain symbolic and as a consequence I cannot calculate the Struve function in order to find the kernel and proceed to the integration. I have the code for the numerical integration but I cannot use it yet.

[Star Strider]

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.