On numerical solution of Burgers' equation by homotopy analysis method

Abstract

In this Letter, we present the Homotopy Analysis Method (shortly HAM) for obtaining the numerical solution of the one-dimensional nonlinear Burgers' equation. The initial approximation can be freely chosen with possible unknown constants which can be determined by imposing the boundary and initial conditions. Convergence of the solution and effects for the method is discussed. The comparison of the HAM results with the Homotopy Perturbation Method (HPM) and the results of [E.1265] are made. The results reveal that HAM is very simple and effective. The HAM contains the auxiliary parameter , which provides us with a simple way to adjust and control the convergence region of solution series. The numerical solutions are compared with the known analytical and some numerical solutions.

Analysis of a New Fractional Model for Damped Bergers’ Equation

“…The damped Burgers' equations has been studied by several researchers such as Babolian and Saeidian [27], Fakhari et al [28], Inc [29], Song and Zhang [30], Peng and Chen [31] and others. The fractional generalization of the damped Bergers' equations is investigated by Esen et al [32].…”

A New Efficient Method for Solving Two-Dimensional Nonlinear System of Burger’s Differential Equations

“…Additionally, like this, it also appears in varied areas of applied mathematics and physics, such as modelling of gas dynamics [1][2][3][4][5]. Recently, many numerical and analytical methods have been used to study the two-dimensional Burger's equation such as the differential transformation method [6], homotopy perturbation method [7], homotopy analysis method [8], variational iteration method [9], Adomian decomposition method [10][11][12], cubic B-spline differential quadrature method [13], finite difference method [14], finite element [15], and local discontinuous Galerkin finite element method [16] and also mathematicians have used transform methods coupled with analytical methods [17][18][19][20][21][22][23][24][25][26][27][28][29][30] to solve PDEs. e Sumudu decomposition method (SDM) is one of these methods, and it has been successfully used to solve intricate problems in engineering mathematics and applied science [31][32][33][34][35].…”

An exploration of quintic Hermite splines to solve Burgers’ equation

“…Burgers' equation has been solved by using a variety of numerical methods such as modified Adomian method [1], homotopy analysis method [17], finite difference scheme [18,20,22,28], automatic differenti-ation [7], finite element method [2], and spline collocation [6,11,19,25,27,31]. Several investigators have solved Burgers' equation by transforming the non-linear term into linear one [18,20,27,28] using Hopf-Cole transformation [10,16].…”