The numerical solution of fredholm integral equations with rapidly varying kernels

摘要

Approximating a solution to the Fredholm integral equation ø(x)=α(x) + ∫ baK(x, y)ø(y) dy by the Nyström method involves some numerical quadrature for approximating the integral, producing a linear system satisfied by approximate function values of ø. This paper discusses the use of generalized product-interpolatory formulas which model ø as one mth-degree polynomial on each subinterval and model K as a (possibly large) sequence of nth-degree polynomials. In cases where K is varying much more rapidly than ø this allows for ø to be sampled much less often than K. Since K is modeled as a sequence of polynomials, its frequent sampling does not require a prohibitive increase in the degree of the interpolating polynomials. Coefficient formulas and examples are given for the (m,n) cases (1,1), (1,2), (2,1) and (2,2).