View Single Post
Sep22-09, 11:20 AM
P: 3
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(x-x'))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(x-x'))g(x')dx'[/tex]
can be written as
I've obtained this results by transorming into a differential equation and solving that.
Please help!!!!!!!
Phys.Org News Partner Science news on
Bees able to spot which flowers offer best rewards before landing
Classic Lewis Carroll character inspires new ecological model
When cooperation counts: Researchers find sperm benefit from grouping together in mice