Numerical solutions of Burgers' equation in a B-polynomial basis

Abstract

A new algorithm for approximating numerical solutions of Burgers' equation in a modified B-polynomial basis is presented. The desired solutions are expanded in terms of a set of continuous polynomials over the entire interval for the spatial part with only the time variable requiring discretization. The algorithm makes use of the Galerkin method to determine the expansion coefficients to construct initial values for expansion coefficients. Matrix formulation is used to construct system of equations in time variable which are solved using a fourth-order Runge–Kutta method. The accuracy of the solutions is dependent on the size of the B-polynomial basis set. The current algorithm is implemented in solving two cases of homogeneous and non-homogeneous boundary conditions. Excellent agreement is found between exact and approximate solutions. This procedure has great potential to be implemented in more complex systems of differential equations where no exact solution is available.

Contemporary review of techniques for the solution of nonlinear Burgers equation

“…In [142], authors used restrictive Taylor approximation classical explicit finite difference method for the solution purpose. Bhatti and Bhatta [143] used B-polynomial basis in the Galerkin formulation of Burgers equation (1) and the matrix formulation is used to construct system of equations in time variable which are solved using a fourth-order Runge-Kutta method. The accuracy of the solutions is dependent on the size of the B-polynomial basis set.…”

Darboux transformations for the time-dependent nonhomogeneous Burgers equation in (1+1) dimensions

“…It is well known [2] that the conventional Darboux transformation (2) of order n factorizes into n first-order Darboux transformations (5). This means…”

“…However, let us study the existence of zeros in the Darboux-transformed Schrödinger solution in more detail. To this end, consider the first-order transformation (5), applied to a solution û of the Schrödinger equation ( 1) for an auxiliary function v 1 . According to the notation in (5), the transformed solution is then called D 1,(v 1 ) ( û).…”

“…Numerical methods can be used to find approximative solutions (see e.g. [5]), but only very few nonhomogeneities are known, for which the nonhomogeneous Burgers equation (NBE) renders integrable. As a well-studied example of such a nonhomogeneity let us mention the linear one (with respect to the spatial variable), see e.g.…”

Algebraic dynamics solution and algebraic dynamics algorithm of Burgers equations

“…Since then, it has been the original model describing interaction of convection and dissipation-flow. In fact, Burgers equation has always been an important equation of fluid dynamics for half a century, and even till now its analytical and numerical solutions are still a subject of study [3][4][5][6][7][8] .…”