A Survey of Some Methods in Numerical DEs 1 A Hodge Podge of Methods for ODEs and PDEs

Method: Collocation Method (a Weighted Residual Method) Given a differential equation L[u] = 0 for u(ξ) with ξ ∈ Q (Q some domain) and boundary conditions B[u] = 0, we seek an approximate solution u(ξ) = w(ξ;α) where α = {αj}j=1 is a set of parameters and B[w] = 0 for all choices of α. We try to determine the α by requiring that the equation be satisfied by w at a set of points S = {ξj}j=1 ⊂ Q. Thus we arrive at a system of N equations in the N unknowns α: L[w(ξj, α)] = 0, j = 1, · · · , N.