Approximation of Burger's Equation Using Sextic B-Spline Galerkin Scheme with Quintic Weight Function

In this study We produce new numerical scheme which rely on sextic B-spline Galerkin method takes with quintic B-spline as a weight function, for solving the Burger's equation, contrasted with exact solution can be done and then we find out a linear stability analysis which is erect on a Fourier (Von Neumann) method.


1-Preface
The Burger's equation first appeared in 1915" [2], where he applied this equation as a sample for the motion of a viscous fluid when the viscosity approaches zero.Later, "Burger investigated various aspects of turbulence and developed a mathematical model illustrating the theory as well as statistical and spectral aspects of the equation and related system" [4], [5], [6].Because of comprehensive work of Burger, it's known as Burger's equation.It play an necessary part in studying different problem for sciences and engineering like a model in fields as wide as heat conduction" [7], "gas dynamics" [12], longitudinal elastic waves in an isotropic solid" [3], number theory [18], and so forth.The Burger's equation is solved analytically and independently for unintentional intinal conditions [11], [7]."Inmany states, these solutions include infinite series which may converge very slowly for small values of viscosity coefficients , which correspond to steep wave fronts in the propagation of the dynamic wave forms.Burgers' equation shows a similar features with Navier-Stokes equation due to the form of the nonlinear convection term and the occurrence of the viscosity term.Before concentrating on the numerical solution of the Navier-Stokes equation, it seems reasonable to first study a simple model of the Burgers' equation.Therefore, the Burgers' equation has been used as a model equation to test the numerical methods in terms of accuracy and stability for the Navier-Stokes equation.Many authors have used a variety of numerical techniques for getting the numerical solution of the Burgers' equation.Numerical difficulties have been come across in the numerical solution of the Burgers' equation with a very small viscosity.Various numerical techniques accompanied with spline functions have been set up for computing the solutions of the Burgers' equation.The nonlinear term makes it more interesting to study as it is one of the few nonlinear partial differential equations that have been solved analytically.

2-Sextic B-Spline Galerkin Scheme with Quintic Weight Function (SBGQWM)
"Consider the one dimensional quasi-linear parabolic differential equation known as Burger's equation given by" [11].(1) where is the coefficient for the kinematic viscosity and subscripts and denote differentiation."The boundary conditions" are selected form If the Galerkin technique is applied to (1) such that is the weight function yields the following integral equation: We consider the mish is a uniform partition of the solution domain a by the knots and , , throughout this paper.
The sextic B-spline ( ) ( ( ) ) which form basis over the solution domain ] at the knots , is defined as [14] ( ) The set of splines ( ( ) ( ) ( ) ( )) represent forms a basis for functions defined over ].The approximate solution ( ) to the exact solution ( ) is Such that are unknown (time-dependent) parameters that will be determined from the weighted residual and boundary conditions.Since each cubic B-spline covers seven elements such that every element ] is covered by seven splines.."Ineach element, we using the following local coordinate transformation" [18] (6) The shape functions in expressions over the element [ ] to the sextic B-spline (4) can be given by:- Using trial function ( 5) and sextic B-spline (4) to determine the value of with its first and second derivatives respectively at the knots ( ) in terms of element parameters as follows:

Weight function ( ) is taken as a quintic B-spline ( ) ( ( ) ), at the knots which form basis over the solution domain
], is defined as [14] ( ) ( ) In each element, using (6), a shape functions of quintic B-spline in terms of over the element [ ] is given by When the Petrov-Galerkin approach is applied to Eq.( 1), Using transformation (6), equation( 3) for the typical element [ ] becomes ([9], [10]) "Where was taken to be a constant over an element to simplify the integral" [8], [17] ∫( ) ( ) and error norms are used for numerical example and comparison is made with results of the paper [6].
"Burgers' equation has the following form of the analytical solution".
Where ( ) .The propagation of the shock is represented by the equation above."The initial shock which is taken when t =1 in Eq." (21) will be observed as time progresses.To make comparison with earlier study [1], computation is done with parameters ), ) and over the domain [0, 1]."Table (1) show a comparison of the exact solution with numerical values to scheme"."Comparisons are presented at time t = 1.7, 2.4 and 3.1 only".The accuracy in the norm obtained is measured as 2.8 × at time t=1.7,2.4 × at time t=2.4 and 1.4 × at time t = 3.1 for the SBGQWM.
The propagation of the shock is visualized at some times in the Figs.(1), "which it is seen that the initial shock becomes steadier as a program runs".At time t = 3.1, the error distribution is drawn over the domain in the Figs.(2), and there appears to be the highest error about the right-hand boundary position.

5-Conclusions
The numerical algorithm based on Sextic B-Spline as trial function Galerkin method and quintic B-splines as weight function and is constructed of both Burgers' equation.The numerical method appear able to producing numerical solution for high accuracy of the solution to the Burgers' equation."Alsowe found that there is not frequently effect of the time-splitting for Burgers' equation on obtainment the numerical solution introduction method.The experimental results of the scheme is a lot more acceptable in comparison with the precedent results [1].
"Therefore we can be concluded that the introduction method is efficient and credible for getting a numerical solution for the partial differential equations.The simulation process is made by using MATLAB 2011 software package.