Classical Methods for FK2

Most of the computational methods for the approximate solution of an integral equation can be regarded as “expansion methods.” Although the quadrature rule solves an FK2 of the form ϕ(x)—λ (Kϕ) (x) = f(x) and yields an approximate solution $$ \tilde{\Phi } $$ , which we take as a vector with functional values $$ \tilde{\phi }({x_{0}}),\tilde{\phi }({x_{1}}),...,\tilde{\phi }({x_{n}}) $$ . These values are used in the Nyström methods, discussed in Section 1.6, to yield the approximation $$ \tilde{\Phi }(x) $$ . We present in this and the next chapter some of these methods including the following methods: quadrature, expansion, collocation, product-integration, and Galerkin, to solve Fredholm equations of the second kind.