Predictor homotopy analysis method and its application to some nonlinear problems

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Abstract

The purpose of this paper is to present a kind of analytical method so-called Predictor homotopy analysis method (PHAM) to predict the multiplicity of the solutions of nonlinear differential equations with boundary conditions. This method is very useful especially for those boundary value problems which admit multiple solutions and furthermore is capable to calculate all branches of the solutions simultaneously. As illustrative examples, the method is checked by the model of mixed convection flows in a vertical channel and a nonlinear model arising in heat transfer which both admit multiple (dual) solutions.

Introduction

Everyone familiar to the analytical and especially numerical methods knows that these methods usually converge to only one solution that is exactly meaning of “convergence”. It is more consequential not to lose any solution of nonlinear boundary value problems in engineering and physical sciences. In other words, the existence of a method capable to answer this question that “how many solutions does given nonlinear differential equation with boundary conditions admit?” could be so important. Based on this important issue the present paper is going to present an analytical method, so-called predictor homotopy analysis method (PHAM), so that it enables us to predict the multiplicity of the solutions which nonlinear boundary value problem admits and furthermore to calculate the multiple solutions analytically at the same time.

The homotopy analysis method (HAM) [1], [2], [3] was first proposed by Liao in 1992 to solve many nonlinear problems. The HAM has been successfully applied to many nonlinear problems, such as 2-dimensional steady slip flow in microchannels [4], calculus of variations [5], Chen system [6], thin film flows of a third order fluid [7]. The powerful analytical method HAM has been already successfully applied to various complicated problems in science and engineering [8], [9], [10], [11], [12] and very recently in [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25]. Also by traditional HAM the multiple solutions of some problems have been obtained [26], [27], [28].

The main idea behind this method is to reconstruct the homotopy analysis method by adding rule of multiplicity of solutions and so-called prescribed parameter. We take advantage of convergence- controller parameter in erudite way in order to achieve this important goal that is anticipating existence of multiple solutions besides obtaining all branches of solutions analytically. The another advantage of this method is to use just one auxiliary linear operator, one auxiliary function and particularly one and the same initial approximation guess to calculate all branches of the solutions simultaneously. Consequently, this method might obtain new unfamiliar class of solutions of nonlinear problems which is of fundamental interest for practical using in science and engineering.

The aforementioned idea has been conjecturally tested on nonlinear model of diffusion and reaction in porous catalysts [29] for the first time. In the present paper, we demonstrate the whole theory of the technique and moreover the legitimacy and reliability of the method is tested by its application to two nonlinear problems arising in mixed convection flows in a vertical channel [30], [31] and heat transfer [32], respectively which both of them admit multiple (dual) solutions that is reason why this models have been chosen to accomplish article’s goal.

Section snippets

The Predictor homotopy analysis method

Consider the nonlinear differential equation:N[u(r)]=0,rΩ,with boundary conditionsBu,un=0,rΓ,where N is general nonlinear operator, B is a boundary operator, and Γ is the boundary of the domain Ω. The crucial step of the predictor homotopy analysis method is based on this fact that the boundary value problem (2.1), (2.2) should be transcribed to equivalent problem so that the conditions (2.2) involves an unknown parameter so-called prescribed parameter δ and are split toBu,δ,un=0,rΓ,u(α)

Mixed convection flows in a vertical channel

The aim of this section is to apply Predictor homotopy analysis method to analyze a kind of model in mixed convection flows namely combined forced and free flow in the fully developed region of a vertical channel with isothermal walls kept at the same temperature [30], [31]. In this model, the fluid properties are assumed to be constant and the viscous dissipation effect is taken into account. The set of governing balance equations for the velocity field is reduced tod4udy4=Ξ16dudy2,with

Nonlinear model arising in heat transfer

Fins are extensively used to enhance the heat transfer between a solid surface and its convective, radiative, or convective radiative surface [33]. Finned surfaces are widely used, for instance, for cooling electric transformers, the cylinders of aircraft engines, and other heat transfer equipment. The temperature distribution of a straight rectangular fin with a power-law temperature dependent surface heat flux can be determined by the solutions of a one-dimensional steady state heat

Conclusions

It is very important not to lose any solution of nonlinear differential equations with boundary conditions in engineering and physical sciences. In this regard, the present paper has introduced a new methodology namely Predictor homotopy analysis method (PHAM) to prevent this so that presented method is not only to predict existence of multiple solutions, but also to calculate all branches of solutions effectively at the same time by only using one initial approximation guess, one auxiliary

Acknowledgements

The respected three anonymous referees have carefully reviewed this paper. As a result of their careful analysis, our paper has been improved. The authors would like to express their thankfulness to them for their helpful constructive comments.

References (34)

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