Numerical Solution of the (2+1)-Dimensional Boussinesq Equation with Initial Condition by Homotopy Perturbation Method

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1 Applied Mathematical Sciences, Vol. 6, 212, no. 12, Numerical Solution of the (2+1)-Dimensional Boussinesq Equation with Initial Condition by Homotopy Perturbation Method a Ghanmi Imed a Faculte des Sciences de Tunis, campus universitaire, 16 Tunis, Tunisia imedghanmi7@yahoo.fr b Boukricha Abderrahmen b Faculte des Sciences de Tunis, campus universitaire, 16 Tunis, Tunisia aboukricha@fst.rnu.tn Abstract In this paper, an application of homotopy perturbation method is to solve the nonlinear (2+1)-dimensional Boussinesq Equation. The analytic of the nonlinear problem is calculated in the form of a series with easily computable components.the numerical solutions are compared with the known analytical solutions.the results prove that HPM is very effective and simple. Keywords: homotopy perturbation method; (2+1)-dimensional Boussinesq Equation 1 Introduction The partial differential equation (PDEs) play an important role in the physical sciences and in engineering fields. In recent years, the studies of the exact solutions for the nonlinear evolution equation have attracted the attention of many mathematicians and physicists [2-7]. Recently, Chen and AL.[9] studied the (2+1)-dimensional Boussinesq Equation by generalized transformation in homogeneous balance method [8] and obtained certain new solitary wave solution; periodic wave solution and the combined formal solitary wave solution and periodic wave solution of the equation. In [1-16], EL-sayed and Kaya are

2 5994 I. Ghanmi and A. Boukricha implemented the Adomian s decomposition for obtaining explicit solutions of the nonlinear partial differential equations. Homotopy perturbation method has been used by many mathematicians and engineers to solve various functional equations. This method was further developed and improved by He and applied to nonlinear oscillators with discontinuities [19],nonlinear wave equation [17], and boundary value problems[15].it can be said that He s homotopy perturbation method is a universal one and is able to solve various kinds of nonlinear functional equations. In this method the solution is considered as the summation of an infinite series which usually converges rapidly to exact solutions. This method continuously deforms the difficult equation under study into a simple equation, easy to solve. For that we will study the (2+1)-dimensional Boussinesq Equation by using the homotopy perturbation method. 2 Basic Idea of homotopy perturbation method The homotopy perturbation method is a combination of the classical perturbation technique and homotopy technique to illustrate HPM consider the following nonlinear differential equation subject to boundary condition A(u) f(r) = ; r Ω (1) B(u, u ) = ;r Γ (2) n where A is a general differential operator, B a boundary operator, f(r) isa known analytic function, Γ is the boundary of the domain Ω and u denotes n differentiation along the normal drawn outward from Ω. The operator A can be generally divided into two parts L and N, where L is linear, whereas N is nonlinear. equation (1) can be rewritten as follows L(u) + N(u) f(r) = (3) By using homotpy technique, we can construct a homotopy ν : Ω [1] R, which satisfies which is equivalent to H(ν, p) =(1 p)[l(ν) L(u )] + p[a(ν) f(r)] = (4) H(ν, p) =L(ν) L(u )+pl(u )+p[n(ν) f(r)] = (5)

3 Homoptopy perturbation method 5995 wherep [, 1] is an embidding parameter and u is an arbitrary initial approximation of equation [1] which satisfies the boundary conditions. When we put p = and p = 1 in equation (4); we obtain H(ν, ) = L(ν) L(u ) = (6) H(ν, 1) = A(ν) f(r) = (7) and the changing process of p from to 1; is just that of H(ν, p) from L(ν) L(u )toa(ν) f(r). In topology this called deformation; L(ν) L() and A(ν) f(r) are called homotopic. Applying the perturbation technique [16], due to the fact that p 1 can be considered as a small parameter, we can assume that the solution of (4) or (5) can be expressed as a series in p, as follows: ν = ν + pν 1 + p 2 ν 2 + p 3 ν (8) When p 1, (4)or (5) corresponds to (3) and becomes the approximate solution of (3): u = lim ν = ν + ν 1 + ν 2 + ν (9) p 1 The series (9) is convergent for most cases; and the rate of convergence depends on A(ν) [17]. The combination of a small parameter(perturbation parameter), with a homotopy is called the HPM as presented in the series (9). The convergence of the series (9) has been proved by He in his paper [18]. 3 Solution of the Boussinesq equation by HPM Consider the following nonlinear differential equation; with the indicated initial conditions: u tt u xx u yy (u 2 ) xx u xxxx = φ(x, y, t) (1) u(x,,t) = f(x, t) (11) u (x,,t) = g(x, t) (12) y for solving this equation by HPM, we rewrite eq(1) as follows: L y (u) N(u) = φ(x, y, t) (13) where the linear operator L y = 2 ; with the inverse operator L 1 y = 2 y (.)dsdz (14)

4 5996 I. Ghanmi and A. Boukricha and N(u) =u tt u xx (u 2 ) xx u xxxx (15) represents the nonlinear term. the inverse operator, we get u(x, y, t) = f(x, t)+yg(x, t)+ φ(x, s, t)dsdz + By HPM L(u) = u(x, y, t) h(x, y, t) = where h(x, y, t) = f(x, t)+yg(x, t)+ Hence, we may choose a convex homotopy such that H(ν, p) =ν(x, y, t) h(x, y, t) p y z N(x, s, t)dsdz (16) φ(x, s, t)dsdz (17) u tt u xx (u 2 ) xx u xxxx dsdz (18) Substituting (8) into (18),and equating the terms with identical powers of p, we have P : ν (x, y, t) = h(x, y, t) (19) P 1 : ν 1 = y z 2 ν (x, s, t) t 2 2 ν (x, s, t) x 2 2 ν (x, s, t) 2 x 2 (2) P 2 : ν 2 = P 3 : ν 3 = 4 ν (x, s, t) x 4 dsdz 2 ν 1 (x, s, t) t 2 2 ν 1 (x, s, t) x ν (x, s, t)ν 1 (x, s, t) x 2 (21) 4 ν 1 (x, s, t) x 4 dsdz 2 ν 2 (x, s, t) t 2 2 ν 2 (x, s, t) x ν (x, s, t)ν 2 (x, s, t) x 2 (22) 2 ν 1 (x, s, t) 2 x 2 4 ν 2 (x, s, t) x 4 dsdz To give a clear overview of the methodology; as an illustrative example,we take (2+1)-dimensional Boussinesq equation described by u tt u xx u yy (u 2 ) xx u xxxx = (23) u(x,,t)=k 6α 2 tanh 2 (αx γt) (24) u y (x,,t)= 12α2 βsech 2 (αx γt) tanh(αx γt) (25)

5 Homoptopy perturbation method 5997 Consequently, solving the boussinesq equation, the first few components of the homotopy perturbation solution for eq.(23)are derived as follows: ν(x, y, t) =k 6 α 2 (tanh (αx+ γt)) 2 12 yα 2 β (sech (αx+ γt)) 2 tanh (αx+ γt) (26) u 1 (x, y, t) = 3α2 y 2 [3α 2 3γ 2 132α 4 +2α 2 cosh(2η) 2 2γ 2 cosh(2η) + 14α 4 cosh(2η) α 2 cosh(4η) +γ 2 cosh(4η) 4α 4 cosh(4η)]sech(η) 6 +α 2 βy 3 [ 9α 2 +9γ α 4 8α 2 cosh(2η)+8γ 2 cosh(2η) 224α 4 cosh(2η)+α 2 cosh(4η) γ 2 cosh(4η) +α 4 cosh(4η)]sech(η) 6 tanh(η) (27) u 2 (x, y, t) = 3α 4 y 2 [3K + 543α 2 +2K cosh(2η) 6α 2 cosh(2η) K cosh(4η)+6α 2 cosh(4rη)]sech(η) 6 + α2 y 4 32 [95α4 19α 2 γ 2 +95γ 4 98α 6 192α 4 β α 4 γ α 8 +86α 4 cosh(2η) 172α 2 γ 2 cosh(2η) +86γ 4 cosh(2η) 1232α 6 cosh(2η) 192α 4 β 2 cosh(2η) +1232α 4 γ 2 cosh(2η) α 8 cosh(2η) 32α 4 cosh(4η)+64α 2 γ 2 cosh(4η) 32γ 4 cosh(4η) +7616α 6 cosh(4η) α 4 β 2 cosh(4η) 7616α 4 γ 2 cosh(4η) α 8 cosh(4η) 22α 4 cosh(6η) +44α 2 γ 2 cosh(6η) 22γ 4 cosh(6η) 944α 6 cosh(6η) 192α 4 β 2 cosh(6η) + 944α 4 γ 2 cosh(6η) 832α 8 cosh(6η)+α 4 cosh(8η) 2α 2 γ 2 cosh(8η) +γ 4 cosh(8η)+8α 6 cosh(8η) 8α 4 γ 2 cosh(8η) +16α 8 cosh(8η)]sech(η) 1 +2α 4 βy 3 [ 9K 21α 2 8K cosh(2η) + 144α 2 cosh(2η)+k cosh(4η) 6α 2 cosh(4η)]sech(η) 6 tanh(η)+ α2 βy 5 8 [ 515α4 + 13α 2 γ 2 515c α 6 62α 4 γ α 8 596α 4 cosh(2η) α 2 γ 2 cosh(2η) 596γ 4 cosh(2η) α 6 cosh(2η) 29792α 4 γ 2 cosh(2η) α 8 cosh(2η) 28α 4 cosh(4η)+56α 2 γ 2 cosh(4η) 28c 4 cosh(4η) 28448α 6 cosh(4η) α 4 γ 2 cosh(4η) α 8 + (52α 4 14α 2 γ 2 +52γ α α 4 γ 2

6 5998 I. Ghanmi and A. Boukricha α 8 ) cosh(6η) (α 4 2α 2 γ 2 + γ 4 +8α 6 8α 4 γ 2 +16α 8 ) cosh(6η)]sech(η) 1 tanh(η) (28) and so on, where η = (αx γt). Therefore, The exact solution of (23) with the initial condition (24) and (25) in closed form will be U(x, y, t) = K 6α 2 tanh(αx + βy γt), where K = 1 γ 4 +8α 4 α 2 β 2 2 α 2 and α, β, γ are arbitrary constants. 4 Numerical results Table 1 The numerical result with t =.5 when α =.1, β =.1,and γ = 1 for the solution of the equation (1) for initial conditions (24)and (25) u(x, y, t) Φ(x, y, t) (x i,y i ) Continuing this process the complete solution u = lim n φ n found by means of n-term approximation φ n = n k= ν k. For this example, for-term approximation φ 4 = ν + ν 1 + ν 2 + ν 3 gives the best convergence. The error, complete(analytical) and the numerical results are given and compared each other in table 1 for some values of x,y,and t =.5. As expected, the numerical solution in table 1 is clearly indicated that how the decomposition scheme obtains efficient result much closer to the actual solutions. It is also worth noting that the advantage of the decomposition methodology shows a fast convergence of the solution. Clearly, the series solution methodolgoy can be applied to much more complicated any other nonlinear problems as well.

7 Homoptopy perturbation method 5999 Figure 1: An approximate numerical solution for φ 3 with t =.5 when α =.1, β =.1,andγ =1 Figure 2: The analytic solution for the (2+1)-dimensional Boussinesq equation with t =.5 when α =.1, β=.1,andγ =1

8 6 I. Ghanmi and A. Boukricha Figure 3: Comparison between numerical solution Φ 3 (x, y, t) and exact solution u(x, y, t) with y =.1,t=.5 when α =.1, β=.1,andγ =1 5 conclusion In this work, we successfully apply the homotopy perturbation method to approximate the solution of nonlinear (2+1)-dimensional Boussinesq equation. It is shown that HPM needs much less computational work compared with traditional method.it is apparently seen that this method is very powerful and an efficient technique for solving different kinds of problems arising in various fields of science and engineering and present a rapid convergence for the solution. Although goal of He s homotopy perturbation method was to find a technique to unify linear and nonlinear, ordinary or partial differential equation for solving initial and boundary value problems. References [1] L.Debtnath, Nonlinear partial differential Equations for Scientist and Engineers,Birkhauser, boston, 1997 [2] T.Hong, Y.Z. Wang, Y.S. Huo,Bogoliubov quasipartcles carried by dark solitonic existations in nonuniform Bose-Einteincondensates, ChinesPhys.Lett.15(1998) [3] T.C.Xiaz, B.Li, H.Q.Zhangx, New explicit and exact solution for the Nizhnik-Novikov-Vesselov equation, Appl.Math.E-Note 1(21)

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