Boundary shape function iterative method for nonlinear second-order boundary value problems with nonlinear boundary conditions

Abstract

Nonlinear boundary conditions are difficult to be fulfilled exactly, when one employs numerical methods to treat a highly nonlinear boundary value problem (NBVP). In this paper, a novel iterative algorithm to solve NBVP involved with two coupled nonlinear boundary conditions at two-end of a unit interval is developed, of which the solution can satisfy the nonlinear boundary conditions automatically. By letting the free function in the boundary shape function (BSF) be a new variable, an initial value problem (IVP) is created from the second-order NBVP. While the initial values of the new variable are given, the terminal values are viewed as unknown parameters to be determined iteratively. Therefore, a very accurate solution for the NBVP with nonlinear boundary conditions can be quickly determined through a few iterations. Some numerical examples confirm the efficiency and accuracy of the proposed iterative scheme, wherein the examples with multiple solutions and unique solution are worked out.(c) 2021 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights reserved.