# Best ways to avoid singularities in kernels when solving integral equations numerically

Take a Fredholm integral equation $$u(x) - \lambda \int_{-1}^{1} K(x,y)u(y) \, \mathrm{d} y = f(x)$$ and discretize it via (say) Gaussian quadrature with nodes $$\{x_j\}$$ and weights $$\{w_j\}$$ to get $$u(x) - \lambda \sum_{j} w_j K(x,x_j)u(x_j) = f(x)$$ It feels natural to use the same set of quadrature nodes over $$f$$ to get a linear system, so $$u(x_k) - \lambda \sum_{j} w_j K(x_k,x_j)u(x_j) = f(x_k)$$ But if $$K$$ is singular, we have a big problem on out hands.

What are the normal methods of dealing with this?

Obviously we could use Gaussian quadrature nodes over $$y$$ and Chebyshev nodes over $$x$$, but that feels a little awkward; are there better ways?

• Look into the locally corrected Nystrom scheme, which is designed specifically for this problem. – sssssssssssss Mar 19 '20 at 11:17
• There is a substantial literature on quadrature in the context of the Boundary Element Method. You might want to look there. – Wolfgang Bangerth Mar 19 '20 at 14:59
• To answer this question, we need to see your equation, not the Fredholm equation in general. – Alex Trounev Mar 19 '20 at 18:13