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Solving second-order nonlinear boundary value problem with nonlinear boundary conditions by an iterative method

Chein-Shan Liu (College of Mechanics and Materials, Hohai University, Nanjing, China and Center of Excellence for Ocean Engineering, Center of Excellence for the Oceans, National Taiwan Ocean University, Keelung, Taiwan)
Jiang-Ren Chang (Department of Systems Engineering and Naval Architecture, National Taiwan Ocean University, Keelung, Taiwan)

Engineering Computations

ISSN: 0264-4401

Article publication date: 19 June 2020

Issue publication date: 27 January 2021

Abstract

Purpose

The purpose of this paper is to solve the second-order nonlinear boundary value problem with nonlinear boundary conditions by an iterative numerical method.

Design/methodology/approach

The authors introduce eigenfunctions as test functions, such that a weak-form integral equation is derived. By expanding the numerical solution in terms of the weighted eigenfunctions and using the orthogonality of eigenfunctions with respect to a weight function, and together with the non-separated/mixed boundary conditions, one can obtain the closed-form expansion coefficients with the aid of Drazin inversion formula.

Findings

When the authors develop the iterative algorithm, removing the time-varying terms as well as the nonlinear terms to the right-hand sides, to solve the nonlinear boundary value problem, it is convergent very fast and also provides very accurate numerical solutions.

Research limitations/implications

Basically, the authors’ strategy for the iterative numerical algorithm is putting the time-varying terms as well as the nonlinear terms on the right-hand sides.

Practical implications

Starting from an initial guess with zero value, the authors used the closed-form formula to quickly generate the new solution, until the convergence is satisfied.

Originality/value

Through the tests by six numerical experiments, the authors have demonstrated that the proposed iterative algorithm is applicable to the highly complex nonlinear boundary value problems with nonlinear boundary conditions. Because the coefficient matrix is set up outside the iterative loop, and due to the property of closed-form expansion coefficients, the presented iterative algorithm is very time saving and converges very fast.

Keywords

Citation

Liu, C.-S. and Chang, J.-R. (2021), "Solving second-order nonlinear boundary value problem with nonlinear boundary conditions by an iterative method", Engineering Computations, Vol. 38 No. 1, pp. 107-130. https://doi.org/10.1108/EC-03-2020-0129

Publisher

:

Emerald Publishing Limited

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