Shooting homotopy analysis method: A fast method to find multiple solutions of nonlinear boundary value problems arising in fluid mechanics

Abstract

Purpose

The purpose of this paper is to present a new method based on the homotopy analysis method (HAM) with the aim of fast searching and calculating multiple solutions of nonlinear boundary value problems (NBVPs).

Design/methodology/approach

A major problem with the previously modified HAM, namely, predictor homotopy analysis method, which is used to predict multiplicity of solutions of NBVPs, is a time-consuming computation of high-order HAM-approximate solutions due to a symbolic variable namely “prescribed parameter”. The proposed new technique which is based on traditional shooting method, and the HAM cuts the dependency on the prescribed parameter.

Findings

To demonstrate the computational efficiency, the mentioned method is implemented on three important nonlinear exactly solvable differential equations, namely, the nonlinear MHD Jeffery–Hamel flow problem, the nonlinear boundary value problem arising in heat transfer and the strongly nonlinear Bratu problem.

Originality/value

The more high-order approximate solutions are computable, multiple solutions are easily searched and discovered and the more accurate solutions can be obtained depending on how nonhomogeneous boundary conditions are transcribed to the homogeneous boundary conditions.

Citation

Ahmad Soltani, L., Shivanian, E. and Ezzati, R. (2017), "Shooting homotopy analysis method: A fast method to find multiple solutions of nonlinear boundary value problems arising in fluid mechanics", Engineering Computations, Vol. 34 No. 2, pp. 471-498. https://doi.org/10.1108/EC-10-2015-0329