A third-derivative two-step block Falkner-type method for solving general second-order boundary-value systems.

Abstract

[EN]In this article, a third derivative continuous 2-step block Falkner-type method for the general solution of second order boundary value problems of ordinary differential equations (ODEs) with different types of boundary conditions is developed. The approaches of collocation and interpolation are adopted to derive the new Falkner-type method, which is then implemented in a block mode to get approximations at all the grid points simultaneously. This method is said to be a global method since it simultaneously produces a solution over the entire interval, although it may also be categorized as a boundary value method (see Brugnano and Trigiante (1998)). The order and the convergence analysis of the proposed method are studied. The new Falkner-type scheme is applied to solve linear and non-linear systems of second-order boundary value problems of ODEs considering different types of boundary conditions. Numerical results obtained through the implementation of the scheme are very much close to the theoretical solution and found favourably compared with various existing methods in the literature.