A high order B-spline collocation method for linear boundary value problems

Abstract

Collocation methods are investigated because of their simplicity and inherent efficiency for applications to linear boundary value problems on [ a , b ] . The objective of the present research is obtaining numerical solution of the boundary value problems for dth order linear boundary value problem by a B-spline collocation method using B-splines of order k and their index of regularity is m, d - 1 ⩽ m ⩽ k - 2 . The collocation points in our method form a strictly increasing sequence of points in [ a , b ] , each interior jth collocation point belongs to the interior of the compact support of corresponding jth B-spline basis element, and the number of B-spline basis elements equals the number of collocation points. The order of accuracy of the proposed method is shown to be optimal. The mathematical properties of this collocation method are less well established , primarily because the order of accuracy depends on the regularity and order of B-spline basis and location of the collocation points. The error analysis is through the Green’s function approach than the matrix approach. We compare the efficiency and accuracy of our method to nodal and orthogonal collocation methods as applied to linear ordinary differential equations with boundary conditions. Our collocation method like Greville collocation method is more convenient than nodal or orthogonal collocation because exactly the correct number of collocation points is available. The Greville and Botella collocation methods are special cases of our collocation method.