Mixed galerkin inite element solution of the homogenous burgers equation
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Date
2015Author
Adenya, K.
Okoya, RO.
Aywa, Shem.
Oganga, OD.
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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
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