Mixed galerkin inite element solution of the homogenous burgers equation

Abstract

Nonlinear partial differential equations arise in a large number of mathematical and engineering problems. Systems of nonlinear partial differential equations have attracted much attention in studying solid state physics, fluid mechanics, chemical, propagation of undular bores in shallow water waves [1], propagation of waves in elastic tube filled with a viscous fluid [2], and plasma physics [3]. Burgers equation is one of the well-known equations in mathematics and physics. This equation has been found to describe various kinds of phenomena such as the mathematical model of turbulence [4] and the approximate theory of flow through a shock wave traveling in a viscous fluid [5]. The Korteweg–de Vries–Burgers (KdV–Burgers) equation is a 1-D generalization of the model description of the density and velocity fields that takes into account pressure forces as well as the viscosity and the dispersion. Several numerical methods are used such as Chebyshev spectral collocation method [6], meshfree interpolation method [7], modified extended backward differentiation formula [8], direct variational methods [9], and so on to solve these equations [10, 11]. In this paper, mixed finite difference [12] and Galerkin methods are used to solve the 1-D, KdV [13], and coupled Burgers equations with interpolating scaling functions (ISFs). Burgers equation in this paper is represented in three types as .E1/ 1-D Burgers equation ut C ˛uux uxx D 0, .x, t/ 2 Œa, b Œ0, T, (1.1) with the initial and boundary conditions u.x, 0/ D f.x/, x 2 Œa, b, (1.2) u.x, t/ D g.t/, .x, t/ 2 Œa, b Œ0, T, (1.3) respectively, where ˛ and are arbitrary constants