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

Aimin Deng, Ji Lin and Chein-Shan Liu

Mathematics and Computers in Simulation (MATCOM), 2022, vol. 194, issue C, 539-551

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.

Keywords: Nonlinear boundary value problem; Nonlinear boundary conditions; Boundary shape function; Iterative algorithm (search for similar items in EconPapers)
Date: 2022
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Persistent link: https://EconPapers.repec.org/RePEc:eee:matcom:v:194:y:2022:i:c:p:539-551

DOI: 10.1016/j.matcom.2021.12.013

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