I have been struggling for a while with a problem I've encountered in my research and I really need help. It is a rather peculiar integral equation
[tex]f_0(x)=S(x)+\int_{x}^b \frac{\Sigma_s}{2}\exp(\Sigma_t(x'x))f_1(x',x'')dx'+\int_{b}^x \frac{\Sigma_s}{2}\exp(\Sigma_t(xx'))f_1(x',x'')dx'[/tex]
in which
[tex]f_1(x',x'')=\int_{x'}^b \frac{\Sigma_s}{2}\exp(\Sigma_t(x''x'))f_0(x'')dx''+\int_{b}^x' \frac{\Sigma_s}{2}\exp(\Sigma_t(x'x''))f_0(x'')dx''[/tex]
and the source [tex]S(x)=C_0+C_1\cosh(\Sigma_tx)[/tex]
It comes from some new theory concerning Monte carlo simulations for neutron transport.
Since I know nothing about the solution function [tex]f_0(x)[/tex], I need an educated guess about its form. If the form is known I can transform into a differential equation and solve for the coefficients. I've tried some stuff but it just wouldn't work.
What may be of help is that the solution to an equation with a single integration which looks like
[tex]g(x)=C_0+\int_{x'}^b \frac{\Sigma_s}{2}\exp(\Sigma_t(x'x))g(x')dx'+\int_{b}^x \frac{\Sigma_s}{2}\exp(\Sigma_t(xx'))g(x')dx'[/tex]
can be written as
[tex]g(x)=A_0+A_1\cosh(\sqrt{\Sigma_t(\Sigma_t\Sigma_s)}[/tex]
I've obtained this results by transorming into a differential equation and solving that.
Please help!!!!!!!
