Multipoint approach to the two-point boundary value problem

Abstract

This paper treats nonlinear, two-point boundary value problems of the form x ̇ − ϑ(x, t) = 0 , in which the Jacobian matrix ϑ x ( x, t) is characterized by large positive eigenvalues. The resulting numerical difficulties are reduced by treating the two-point boundary value problem as a multipoint boundary value problem. A totally finite-difference approach is employed, thus bypassing the integration of the nonlinear equations, which characterizes shooting methods. The approach employed consists of extending to multipoint boundary value problems the modified-quasilinearization method developed by Miele and lyer for two-point boundary value problems. Basic to the method is the consideration of the performance index P, which measures the cumulative error in the differential equations, the boundary conditions, and the interface conditions. A modified-quasilinearization algorithm is generated by requiring the first variation of the performance index δP to be negative. This algorithm differs from the ordinary-quasilinearization algorithm because of the inclusion of the scaling factor or stepsize α in the system of variations. The main property of the modified-quasilinearization algorithm is the descent property: if the stepsize α is sufficiently small, the reduction in P is guaranteed. Convergence to the desired solution is achieved when the inequality P ⩽ ϵ is met, where ϵ is a small, preselected number. The variations per unit stepsize Δx(t) α = A(t) are governed by a system of mn nonhomogeneous, linear differential equations subjected to p initial conditions, q final conditions, and ( m − 1) n interface conditions, with p + q = n, where n is the dimension of the vector x and m is the number of subintervals. Therefore, the total number of boundary conditions and interface conditions is mn. The above system is solved employing the method of particular solutions: m( n + 1) particular solutions are combined linearly, and the coefficients of the combination are determined so that the linear system is satisfied. Two numerical examples are presented, one dealing with a linear system and one dealing with a nonlinear system. The examples illustrate the effectiveness as well as the rapidity of convergence of the present method.