- #1
LeonhardEuler
Gold Member
- 860
- 1
Hello Everyone. An interesting equation has come in my thesis research, and I was wondering whether anyone had any useful information about it. It is this equation:
[tex]\int_{a_1}^{b_1}...\int_{a_n}^{b_n}P(x_1,...x_n)K(x_1,...x_n,s_1...s_n)dx_1...dx_n=C[/tex]
K is a known function of the x's and s's. C is an unknown constant. P is a probability distribution and so subject to
[tex]\int_{a_1}^{b_1}...\int_{a_n}^{b_n}P(x_1,...x_n)dx_1...dx=1[/tex]
[tex]P(x_1,...x_n)\ge 0[/tex]
The goal is to find P. I found this Wikipedia page:
http://en.wikipedia.org/wiki/Fredholm_integral_equation
So I see that this is a Fredholm integral equation of the first kind. However, none of the theorems they present have any clear relevance to helping solve this equation, and there is nothing about how to impose the probability conditions.
It would be great if there was a general method for a numerical solution of these equations, and it would be good to know about analytic solutions in certain cases. A general analytic solution is probably too much to hope for. A way to transform this to a differential equation would also be good, since I know more about those.
[tex]\int_{a_1}^{b_1}...\int_{a_n}^{b_n}P(x_1,...x_n)K(x_1,...x_n,s_1...s_n)dx_1...dx_n=C[/tex]
K is a known function of the x's and s's. C is an unknown constant. P is a probability distribution and so subject to
[tex]\int_{a_1}^{b_1}...\int_{a_n}^{b_n}P(x_1,...x_n)dx_1...dx=1[/tex]
[tex]P(x_1,...x_n)\ge 0[/tex]
The goal is to find P. I found this Wikipedia page:
http://en.wikipedia.org/wiki/Fredholm_integral_equation
So I see that this is a Fredholm integral equation of the first kind. However, none of the theorems they present have any clear relevance to helping solve this equation, and there is nothing about how to impose the probability conditions.
It would be great if there was a general method for a numerical solution of these equations, and it would be good to know about analytic solutions in certain cases. A general analytic solution is probably too much to hope for. A way to transform this to a differential equation would also be good, since I know more about those.