Using Bernoulli Equation to Solve Burger's Equation

Transcription

1 Vol.() Using Bernoulli Equation to Solve Burger's Equation Saad N. AL-Azawi* Muna Saleh * Received, December, Accepted, June, Abstract: In this paper we find the exact solution of Burger's equation after reducing it to Bernoulli equation. We compare this solution with that given by Kaya where he used Adomian decomposition method, the solution given by chakrone where he used the Variation iteration method (VIM)and the solution given by Eq()in the paper of M. Javidi. We notice that our solution is better than their solutions. Key words: Burger's equation, kdv equation, PDF.. Introduction: We consider the following Burger's equation is positive parameter This equation arises in various areas of science. Equation (.) has been used in the study of the propagation through liquid-filled elastic tube[].and description for shallow water waves on a viscous fluid [].Equation(.)is used as a model for traffic flow [].Many Researchers had proposed various kinds of exact and numerical solution [,,6,7,8,9,] where they used Adomian decomposition method, variation iteration method, Galerkin method.. Derivation of Burger's equation.[9] We recall the differential form of the nonlinear conservation equation and allow a jump discontinuity for p and q. In many physical problems of interest it would be a better approximation to assume that q is a function of the density gradient as well as p.a simple model is to take is a positive constant.substituting (.) into (.), we obtain the nonlinear diffusion equation, We multiply (.) by Hence: And therefore: to obtain To investigate the nature of the discontinuous solution or shock waves, we assume a function relation If is a quadratic function in,then is linear in p, and. Consequently (.) * Department of Mathematics. College of science for women. Baghdad University

2 Vol.() As a simple model of turbulence, is replaced by the fluid velocity field to obtain the well-known Burgers equation as i.e. is the kinematic viscosity.thus,the Burgers equation is a balance between time evaluation, nonlinearity and diffusion.. Solution of Burger's equation. To solve Burger's equation (.), we can assume the solution c is an arbitrary constant. Hence, from (.) we get Now, let [In order to eliminate c from equation (.)] substituting (.6) into (.) i.e. - is called the travelling wave solution-substituting (.) into (.) Let to get Rewrite (.) as a system of first order O.D.E s Let Then (.7) represent the velocity and a acceleration respectively. Then Notice that equation (.9) represents Bernoulli equation. To solve it Suppose Substituting (.) into (.9) This system has an infinite number of equilibrium points which is axis To solve (.) divide the two equations to get We get

3 This equation is first-order linear differential equation, its integrating factor is And its solution is gives by Vol.() BC: BC: Clearly u (, ) = (stationary case) BC so, is an arbitrary constant BC From (.), we get the solution of Bernoulli equation (.9) By using (.6) and (.) Now, from (.6) so the traveling wave solution (.) Therefore, Remember that from (.8) k is given in (.8) and an arbitrary constant where are arbitrary constants. And. The solution of Burger's equation with Dirichlet conditions. From (.) the solution of Burger's equation is Dirichlet conditions are is And from (.) From which we get From(.) and (.)we determine After determining these constants we

4 find the solution of Burger's equation with Dirichlet conditions. As a special case let then from (.8) we get Vol.() solution, while our solution is exact solution. The solution obtained by Javidi And from (.) we get Or Case : Case: when, A=, B= ) and C=,k=., c= Clearly our solution is easier than his solution. in the solution function α=-.7, =. The graphs of (.),(.6) are in figures ()and() respectively. The two shapes have the same qualitative behavior but quantitavely different. where t=[:.:.],x=[-::] This figure classifies the wave at the beginning of the earth quake..comparison. The solution of this equation obtained by Kaya where is not easier than our solution. The solution obtained by Omar is approximated

5 Case : - Case: - Fig.-- Fig.-- References:. R.S.Jonson, 97.A Non Linear Equation Incorporating Damping and Disper sion.j.phys.mech.()9-6.. R.S.Jonson, 97.Shallow Water Wave in Viscous Fluid-the Undular Bore.phys- Fluids () Vol.(). Md Abdur Rab,.Some Travelling Wave Solutions of KdV- Burgers Equation.Int. Journal of Math. Analysis, Vol. 6(): - 6. Dogan Kaya,.An application of the decomposition method for the KdVB Equation.Applied Mathematics and Computation () Omar Chakrone, Okacha Diyer, Driss Sbibih.Improved numerical solution of Burger's equation.bol.soc.paran.mat. (s)v.8 ():9-6. J.Biazar and H.H.Ghazvini, 9.Exact and numerical for non - linear Burger's Equation by VIM. Mathematical and Computer Modelling, J.H.He, 7.Variational iteration method - Some recent results and new inter-pretations. Mathematical Computer. 8. X.H.Zhang, J.Ouyang, L.Zhang, 9.Element - free characteristic Galerkin method for Buger's equation. Engineering Analysis with Boundary Elements, Jawad Kadhim, 7.Non-Classical Variation Formulation Approach for solving one-dimensional Non linear Partial Differential Burger's Problem.M.S.C thesis, Baghdad University. M. Javidi, 6.A numerical solution of Burgers equation based on modied extended BDF scheme. International Mathematical Forum, ( )6-7 استخدام معادلة برنو لي لحل معادلة بيركر سعد ناجي العزاوي* *قسم الرياضيات كلية العلوم للبنات جامعة بغداد. منى صالح* الخالصة: حصلنا من بحثنا هذا على الحل المضبوط )Exact( لمعادلة بيركر بعد تحويلها إلى معادلة برنو لي وتمت مقارنة هذا الحل مع حل " Kaya "الذي استخدم طريقة تجزئة ادومين وحل "Chakrone" الذي أستخدم طريقة التغاير المتكرر وحل "Javidi" المعطى بالعالقة ) (وتبين ان حلنا هذا افضل من حلولهم للبساطه الصيغة