How to Apply the Fredholm Method with a Sum of Separable Kernels?

[Dustinsfl]

How does one use the Fredholm Method when we have a sum of separable kernels?
\[
f(x) = 1 + \lambda\int_0^1(xy + x^3y^2)f(y)dy
\]
Do we or can we just say \(K(x, y) = xy + x^3y^2\) and proceed normally finding the Fredholm Resolvent? Or do we have to use the sum of separable kernels method with the Fredholm method?

If the answer we have to use the sum of separable kernels method with the Fredholm method, how is this done?

 

[Aaron Bex]

Yes, you must use the sum of separable kernels method with the Fredholm method in order to solve this problem. The steps for doing this are as follows:

1. Separate the given function into a sum of two separate functions, \(f_1(x) = x\) and \(f_2(x) = x^3\).

2. Generate the corresponding kernels, \(K_1(x, y) = f_1(x)f_2(y)\) and \(K_2(x, y) = f_2(x)f_2(y)\).

3. Apply the Fredholm Resolvent to each of the separated kernels, i.e.,
\[
R_1(\lambda) = \int_0^1 K_1(x, y)f(y) dy
\]
and
\[
R_2(\lambda) = \int_0^1 K_2(x, y)f(y) dy.
\]

4. Sum up the results from the two Fredholm Resolvents, i.e.,
\[
f(x) = 1 + \lambda\left(R_1(\lambda) + R_2(\lambda)\right).
\]
This is the solution using the Fredholm Method with the sum of separable kernels.