Solving Integral Equation with Eigenfunctions

[eljose79]

i am trying to solve the integral equation

g(x)=Int(0,infinite)f(t)st/exp(st)-1)) the Kernel is

K(s,t)=st/(exp(st)-1) so K(s,t)=K(t,s) is symmetric..so their eigenfunctions will be orthogonal and their eignevalues real..but i do not kow if f(t) belongs to L**2 so we could i solve it using a series of eigenfunctions?...thanks.

what otgher method to approach the solutions are known?..

 

[matt grime]

1. the left hand side is a function of x. there is no x on the righthand side
2. what is the integral with respect to? ds, dt, d(st)?

 

[tacman]

There are several methods that can be used to solve integral equations with eigenfunctions. One approach is to use the method of separation of variables, where the solution is expressed as a series of eigenfunctions and their corresponding coefficients are determined by solving a system of equations. This method is often used when the kernel is separable, meaning it can be expressed as a product of functions of each variable.

Another approach is to use the Fredholm integral equation theorem, which states that the solution to a certain class of integral equations can be found by transforming the equation into a matrix equation and then solving for the eigenvalues and eigenvectors of the resulting matrix. This method is useful when the kernel is not separable.

In terms of determining if f(t) belongs to L**2, this can be checked by calculating the integral of |f(t)|^2 over the interval [0,∞) and seeing if it converges. If it does converge, then f(t) belongs to L**2 and the method of eigenfunctions can be used to solve the integral equation.

Other methods for solving integral equations include using numerical techniques such as the method of moments, collocation methods, and Galerkin methods. These methods involve approximating the integral equation with a finite number of equations and then solving the resulting system.

In conclusion, there are various methods that can be used to solve integral equations with eigenfunctions, depending on the properties of the kernel and the function f(t). It is important to choose the most suitable method for each specific problem in order to obtain an accurate and efficient solution.