Can the Separation of Variables Technique Solve This Integral Equation?

[Karlisbad]

Let be the integral equation:

$$g(s)g(p)g(u)= \int_{0}^{\infty}dx\int_{0}^{\infty}dy\int_{0}^{\infty}dzK(sx)K(py)K(uz)f(x,y,z)$$

then my question is if we could "seek" for a solution in the form:

$$f(x,y,z)=A(x)A(y)A(z)$$ where the function A satsify (for x y and z) the integral equation:

$$g(s)=\int_{0}^{\infty}dxK(xs)A(x)$$ 8and the same for the other)

¿is this approach good?

 

[HallsofIvy]

Yes, you can- though the general solution may be a linear combination of such solutions.

 

[tacman]

The approach of seeking a solution in the form of f(x,y,z) = A(x)A(y)A(z) is known as "separation of variables" and it is a commonly used technique in solving partial differential equations. In this case, it can be applied to the integral equation given above.

This approach can be effective in simplifying the problem and finding a solution, as it reduces the problem to solving three separate integral equations for A(x), A(y), and A(z). However, it is not always guaranteed to work and may not always lead to a unique solution.

It is important to note that this technique relies on the assumption that the solution can be expressed as a product of functions of each variable separately. This may not always be the case, especially if the integral equation is highly nonlinear.

In summary, while seeking a solution in the form of f(x,y,z) = A(x)A(y)A(z) can be a useful approach, it is not always a guaranteed method for solving integral equations. It is important to carefully consider the problem and determine if this technique is appropriate before applying it.