Analytical solution for nonlinear Gas Dynamic equation by Homotopy Analysis Method

Transcription

1 Available at Appl. Appl. Math. ISSN: Vol. 4, Issue (June 29) pp (Previously, Vol. 4, No. ) Applications and Applied Matheatics: An International Journal (AAM) Analytical solution for nonlinear Gas Dynaic equation by Hootopy Analysis Method Hossein Jafari, Changbu Chun 2, S. Seifi, M. Saeidy Departent of Matheatics University of Mazandaran Babolsar, Iran jafari@uz.ac.ir 2 Departent of Matheatics Sungkyunkwan University Suwon , Korea cbchun@skku.edu Received: Septeber 3, 28; Accepted: January 6, 29 Abstract In this paper, the Hootopy Analysis Method (HAM) is used to ipleent the hoogeneous gas dynaic equation. The analytical solution of this equation is calculated in for of a series with easily coputable coponents. Keywords: Nonlinear Gas Dynaic Equation; Hootopy Analysis Method; Partial Differential Equation MSC (2) No.: 47J3; 35G25. Introduction Recently various iterative ethods are eployed for the nuerical and analytical solution of partial differential equation. In this paper, the Hootopy analysis ethod (992) is applied to solve a kind of partial differential equations. 49

2 5 Jafari el al. In 992, Liao eployed the basic ideas of the hootopy in topology to propose a general analytic ethod for nonlinear probles, naely Hootopy Analysis Method (HAM), [Liao (992, 995, 22, 23, 25)]. This ethod has been successfully applied to solve any types of nonlinear probles see Ayub (24a, 24b), Jafari (29) and Liao (24c, 25a). The HAM offers certain advantages over routine nuerical ethods. Nuerical ethods use discretization which gives rise to rounding off errors causing loss of accuracy, and requires large coputer eory and tie. This coputational ethod yields analytical solutions and has certain advantages over standard nuerical ethods. The HAM ethod is better since it does not involve discretization of the variables and hence is free fro rounding off errors and does not require large coputer eory or tie. The paper has been organized as follows. In Section 2 the Hootopy Analysis Method is described. In Section 3 HAM is applied for nonlinear hoogeneous gas dynaics equation. Discussion and conclusions are presented in Section Basic idea of HAM Consider the following differential equation Nurt [ (, )], () where N is a nonlinear operator, r and t are independent variables, u(r, t) is an unknown function, respectively. For siplicity, we ignore all boundary or initial conditions, which can be treated in the siilar way. By eans of generalizing the traditional hootopy ethod, Liao (23) constructs the so called zero order deforation equation (- p) L[ ( r, t; p) - u ( r, t)] phh( r, t) N[ ( r, t; p)], (2) where p [,] is the ebedding paraeter, is a nonzero auxiliary paraeter, H(,) r t is non zero auxiliary function, L is an auxiliary linear operator, u ( r, t ) is an initial guess of urt (,), urt (,: p ) is a unknown function, respectively. It is iportant, that one has great freedo to choose auxiliary things in HAM. Obviously, when p = and p =, it holds (,;) rt u(,), rt (,;) xt urt (,), (3) respectively. Thus, as p increases fro to, the solution ( x, t; p) varies fro the initial guesses u ( r, t) to the solution urt (,). Expanding ( x, t; p) in Taylor series with respect to p, we have (,; rt p) u (,) rt u (,) rt p, (4)

3 AAM: Intern. J., Vol. 4, Issue (June 29) [Previously, Vol. 4, No. ] 5 where ( rt, ; p ) u(,) r t.! p p (5) If the auxiliary linear operator, the initial guess, the auxiliary paraeter h, and the auxiliary function are so properly chosen, the series (4) converges at p =, then we have ( r, t) u ( r, t) u ( r, t) u. (6) Define the vector un u, u,..., u n. Differentiating equation (2) ties with respect to the ebedding paraeter p and then setting th p = and finally dividing the by!, we obtain the order deforation equation Lu [ u ] (, ) ( ), hh rt u (7) and where N [ ( r, t ; p )] ( u ), ( )! p,,,. (9) Applying L on both side of equation (7), we get u (,) r t u (,) [ (,) ( )] r t hl H r t x. () In this way, it is easily to obtain u for, at th M order, we have M uxt (, ) u( xt, ). () When, M we get an accurate approxiation of the original equation (). For the convergence of the above ethod we refer the reader to Liao (23). If equation () adits unique solution, then this ethod will produce the unique solution. If equation () does not possess unique solution, the HAM will give a solution aong any other (possible) solutions.

4 52 Jafari el al. 3. Applying HAM In this section, we apply this ethod for solving the nonlinear gas dynaic equation. Exaple Consider the hoogeneous differential [Evans (22)] 2 u u 2 u u x t ( ) ;,, t 2 x (2) with initial conditions ux (,) ae x. To solve the equation (2) by eans of hootopy analysis ethod, according to the initial conditions denoted in equation (2), it is natural to choose u ( x, t) ae x. (3) We choose the linear operator ( x, t; p) L[ ( x, t; p)], t (4) with the property Lc []. Where c is constant. We now define a nonlinear operator as N[(,; x t p)] t (,; x t p) (,; x t p) x(,; x t p) (,; x t p)( (,; x t p)). (5) Using above definition, with assuption H( x, t). equation We construct the zeroth order deforation ( p)[(,; L r t p) u (,)] r t ph(,) r t N[(,; r t p)], obviously, when p and p, (,;) x t u (,), x t (, x y;) u(,), x t (7) Thus, we obtain the th order deforation equations Lu [ u ] h ( u ), (8) where

5 AAM: Intern. J., Vol. 4, Issue (June 29) [Previously, Vol. 4, No. ] 53 u uu ( ) i i i u u u u t 2 x i i i. Now, the solution of the th order deforation equation (8) u ( x, t ) u (, ) [ ( )]. x t hl x (9) Finally, we have urt (,) u (,) (,). xt u xt Fro equations (3) and (9) and subject to initial condition u(,) x,, we obtain Hence, u (, ), x t ae x (, ) u x, x t ae ht x 2 2 x 2 x u2( x, t) ae t h ae th ae th, 2. uxt (, ) u( xt, ) u( xt, ) x th 4 3th th th ae (4th 6th 3t h 4th 4 t h th...) When h =- we have i (,) x t t t ( ) x t u x t ae t ae ae tx, i i! which is the exact solution of equation (2).

6 54 Jafari el al. 4. Conclusion In this paper, the Hootopy Analysis Method has been applied to study the nonlinear gas dynaic equation. The explicit series solutions gas dynaics equation are obtained, which are the sae as those results given by Adoian decoposition ethod for h. This accords with the conclusion that the hootopy analysis ethod logically contains the Adoian decoposition ethod in other words the ADM is only a special case of the HAM [Liao (24c, 25)]. It is worth pointing out that this ethod presents a rapid convergence for the solutions. In conclusion, HAM provides accurate nuerical solution for nonlinear probles in coparison with other ethods. It also does not require large coputer eory and discretization of the variables t and x. The results show that HAM is powerful atheatical tool for solving nonlinear partial differential equations. Matheatica has been used for coputations in this paper. REFERENCES Ayub, M., Rasheed, A., Hayat, T. (23). Exact flow of a third grade fluid past a porous plate using hootopy analysis ethod. Int. J. Eng Sci, 4 pp Cang, J., Tan, Y., Xu, H., Liao, S.J. (27). Series solutions of non-linear Riccati differential equations with fractional order. Chaos Solitons Fractals, [in press], Doi:./J.chaos Evans, J., Bulut, H. (22). A new approach to the Gas dynaics equation: an application of the decoposition ethod. Intern. J. Coputer Math., Vol. 79(7), pp Hayat, T., Khan, M., Ayub, M. (24). On the explicit analytic solutions of an Oldroyd 6- constant fluid. Int. J. Eng Sci, 42, pp Hayat, T., Khan M., Ayub, M. (24). Couette and Poiseuille flows of an Oldroyd 6-constant fluid with agnetic field. J. Math. Anal. Appl., 298 pp Jafari, H., Seifi, S. (29). Hootopy analysis ethod for solving linear and nonlinear fractional diffusionwave equation. Coun. Nonlinear Sci. Nuer Siulat, 4 pp Liao, S.J. (992). The proposed hootopy analysis technique for the solution of nonlinear probles, Ph.D. Thesis, Shanghai Jiao Tong University, 992. Liao, S.J. (995). An approxiate solution technique which does not depend upon sall paraeters: a special exaple. Int. J. Nonlinear Mech, 3, Liao, S.J. (23). Beyond Perturbation: Introduction to the Hootopy Analysis Method. CRC Press, Boca Raton: Chapan & Hall; Liao, S.J. (24). On the hootopy analysis ethod for nonlinear probles. Appl Math Coput, 47, Liao, S.J. (25). Coparison between the hootopy analysis ethod and hootopy perturbation ethod. Appl. Math. Coput, 69, pp Liao, S.J. (25a). A new branch of solutions of boundary layer flows over an ipereable stretched plate. Int. J. Heat Mass Transfer,48 pp Liao, S.J. (24c). Explicit analytic solution for siilarity boundary layer equations. Int. J. Heat Mass Transfer, 47, 75-8.