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I am looking to confirm if what I am doing is the proper procedure. I writing a program to discretely solve a Homogeneous Fredholm Equation of Second Kind that is set up as follows:

$ \int \limits_{0}^{t_{0}} {K(\tau_{1},\tau_{2}) f(\tau_{2}) \mathrm{d}\tau_{2}} = \lambda f(\tau_{1})$

There is a going to be a symmetric Kernel function, $K(\tau_{1},\tau_{2})$, involved in this set up. I realize to discretely solve this equation the Kernel can be formed into a two dimensional matrix, and hence forming this into a typical matrix eigenvalue, eigenvector problem. But in order to properly take into account the integral in the equation, I would need to additionally use some numerical methods. For example, if I wanted to proceed with the integration by using the Trapezoidal method, would I need to multiply the kernel matrix by $\frac{t_{0}}{2N}$, with N+1 being the number of discrete samples I have, and multiply all the inside terms by 2? In the end the original symmetric kernel will not be symmetric do to the coefficients that it needs to be multiplied by?

If anybody has any ideas how to discrete solve this problem, any help would be greatly appreciated. Thank you very much.

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