# Double Fredholm integral equation

A Fredholm integral equation of the second kind with separable integral kernel is may be solved as

$$y(x)=f(x)+ \displaystyle\int_a^b K(x,t)y(t)dt$$

then,

$$\displaystyle\int_a^b y(x)N_i(x)dx = \displaystyle\int_a^b f(x)N_i(x)dx + \lambda \displaystyle\sum_{j=1}^nc_j \displaystyle\int_a^b M_j(x)N_i(x)dx$$

and the idea is to solve the system of equations.

but if I find a system of the style $$\displaystyle\int_a^b y(x)N_i(x)dx - \displaystyle\int_a^b f(x)N_i(x)dx - \lambda_1 \displaystyle\sum_{j=1}^nc_j \displaystyle\int_a^b M_j(x)N_i(x)dx = \displaystyle\int_a^b y(x)K_i(x)dx - \displaystyle\int_a^b f(x)K_i(x)dx - \lambda_2 \displaystyle\sum_{j=1}^nc_j \displaystyle\int_a^b H_j(x)K_i(x)dx$$ It shouldn't have a solution because the number of unknowns doubles, right?

Actually I have not been able to write in matrix terms, and I think that it is not possible. I could ask for $$K$$ and $$N$$ to be orthogonal, but that doesn't help along the way either.

• What is the explicit expression of $K(x,t)$? May 26 at 5:59