The Method of Multi-Scale Asymptotic Expansions and Its Corresponding Finite Element Algorithm for the Problem of Heat Exchange in Composite Plane Wall

Abstract

In this paper, we discuss the problem of heat exchange in composite plane wall whose heat transmitting coefficient is not a locally periodic function. For that problem, we present a method of multiscale asymptotic expansions and then propose its corresponding FE algorithm. Of course, we give comprehensive theoretical analysis for the FE algorithm. Finally, we use numerical experiments to investigate them.

Comparison of optimal homotopy analysis method and fractional homotopy analysis transform method for the dynamical analysis of fractional order optical solitons

“…Suppose the auxiliary linear operator, the initial guess, the auxiliary parameter h, and the function H(x, t) are considered such that equation (30) is convergent at r = 1, we obtain…”

“…In the past few decades, semi-analytical techniques such as perturbation, iteration, and variational parameter techniques received operational attention to find approximate solutions of the problems arising in the field of nano-electromechanical systems, textile engineering, and problems defined by different weak as well as strong nonlinear equations. 1,[27][28][29] More particularly, a method of multiscale asymptotic expansion and its corresponding finite element algorithm is discussed by He et al 30 Recently, He's max-min method is applied to solve the strong nonlinear oscillators and higher order Duffing equation. 31 The optimal homotopy analysis method (OHAM), which is an asymptotic method, is effective to obtain the series solution of nonlinear partial differential equations and is a suitable approach to control convergence of approximate solution.…”