Handling the fractional Boussinesq-like equation by fractional variational iteration method

Transcription

1 6 ¹ 5 Jun., COMMUN. APPL. MATH. COMPUT. Vol.5 No. Å 6-633()-46-7 Handling the fractional Boussinesq-like equation by fractional variational iteration method GU Jia-lei, XIA Tie-cheng (College of Sciences, Shanghai University, Shanghai 444, China) Abstract The fractional variational iteration method (FVIM) is an efficient tool to solve fractional differential equations. In this paper, the time-fractional Boussinesq-like equation is solved by the FVIM and a single-soliton solution of Boussinesq-like B(, ) equation is obtained as a special case. Key words modified Riemann-Liouville derivative; fractional variational iteration method; Boussinesq-like equation Mathematics Subject Classification 35Q5 Chinese Library Classification O 75.9 ÃÎÇÃ À ËÃÎÇÊ Boussinesq Á ( Å 444) «(FVIM)» ½ «³ µ Boussinesq ± ¼ º ² Æ Boussinesq B(.) ÄÆ ¾ Riemann-Liouville «Boussinesq Í ÂÉÅ 35Q5 ÏÂÉÅ O 75.9 Ì A Introduction The fractional differential equations have been concerned by a great deal of researchers in recent years since they are widely used in physics and engineering such as fractional Brow- Received: -3-5; Revised: -4-7 Project supported by the Natural Science Foundation of China (6747), the Natural Science Foundation of Shanghai (9ZR48), the Shanghai Leading Academic Discipline Project (J5) Corresponding author: Xia Tie-cheng(96 ), Male, Ph.D., Prof., soliton theory and integrable systems. xiatc@yahoo.com.cn

2 GU Jia-lei, et al: Handling the fractional Boussinesq-like equation 47 nian motion [-], non-random fractional optimal control [3] and so on [4]. Recently, Wu [5-6] proposed a fractional functional and extended the variational iteration method (FVIM) [7] to search the approximative solution of fractional differential equations. It is also successfully applied to solve differernt kind of equations [8-9]. The Boussinesq-like equations have been studied by many authors. Zhu [9] discussed some exact special solutions with solitary patterns and Yan [] derived some new similarity reductions and compacton solutions for Boussinesq-like equations. The main objective of the paper is to obtain approximate solution of the time-fractional Boussinesq-like equation with the FVIM. We note that the fractional derivatives are considered as Jumarie s modified derivative in this paper. In the following, we shall use at will and for convenience, the notations and D α f(x) = f (α) (x) = dα f(x) dx α D α xf(x, y) = f (α) x (x, y) = α f(x, y) x α = α x f(x, y) for the fractional derivative and the fractional partial derivative, respectively. Modified Riemann-Liouville derivative Definition. Let f : R R, x f(x), denote a continuous (but not necessarily differentiable) function. Then its fractional derivative of order α is defined by the expression [] f (α) (x) = Γ( α) (x ξ) α (f(ξ) f())dξ, α <. (.) For positive α, one will set ( f (α) (x) = f (x)) (α ), < α < (.) and f (α) (x) = d = Γ( α) dx (x ξ) α (f(ξ) f())dξ (.3) ( f (α n) (x)) (n), n α < n +, n. (.4) The modified Riemann-Liouville derivative has following properties [] : (i) The αth derivative of a constant is zero. (ii) product law Let u(x), v(x) be non-differentiable functions, we have the fractional Leibniz (u(x)v(x)) (α) = u (α) (x)v(x) + u(x)v (α) (x), < α.

3 48 ¹ 5 (iii) (x γ ) (α) = Γ(+γ) Γ(+γ α) xγ α, γ >. The integral with respect to (dx) α is defined as the solution of the fractional differential equation dy = f(x)(dx) α, x, < α, y() = y, (.5) which solution is provided by the following result: Lemma. Let f denote a continuous function. The solution of the Eq. (.5) is f(ξ)(dξ) α = α The fractional integration by part formula [] b a (x ξ) α f(ξ)dξ. (.6) b u (α) (x)v(x)(dx) α = Γ( + α)[u(x)v(x)] b a u(x)v (α) (x)(dx) α. (.7) a 3 Basic ideas of variational iteration method Consider the differential equation of the form Lu(x, t) + Nu(x, t) = f(x, t), (3.) where L is a linear operator, N is a nonlinear operator and f(x, t) is a known analytical function. According to the variational iteration method [7], we can construct the following correction functional u n+ (x, t) = u n (x, t) + λ(ξ, t)(lu n (ξ, t) + Nũ n (ξ, t) f(ξ, t)) dξ, (3.) where λ is a general Lagrange multiplier which can be identified via variational theory, ũ n is considered as restricted variation, i.e. δũ n =. After finding the Lagrange multiplier, the successive approximation u n (x, t), n of the solution u(x, t) will be readily obtained by using any selective function u (x, t). Consequently, the exact solution may be obtained as u(x, t) = lim n u n(x, t). (3.3) 4 Application to time-fractional Boussinesq-like equation As an example we consider the time-fractional Boussinesq-like equation for < α : D α t u + (u ) xx (u ) xxxx =, x, t >, (4.)

4 GU Jia-lei, et al: Handling the fractional Boussinesq-like equation 49 where Dt α u = Dt αdα t u, subject to the boundary conditions: u(x, ) = 4 ( x) 3 sinh 4 and Dt α u(x, ) = 3 sinh( x). (4.) The iteration formula for Eq. (4.) can be constructed as follows u n+ (x, t) = u n (x, t) + Γ( + α) λ(τ, t) ( D α τ u n(x, τ) + (ũ n (x, τ)) xx (ũ n (x, τ)) xxxx) (dτ) α, n =,,.(4.3) Direct calculation leads to δu n+ (x, t) = δu n (x, t) + Γ( + α) δ λ(τ, t)d α τ u n(x, τ)(dτ) α = δu n + λδu (α) n,τ τ=t λ (α) τ δu τ=t n + Γ( + α) λ (α) τ δu n (dτ) α. (4.4) Setting the coefficients of δu n and δu (α) n to zero yields the following stationary conditions: λ (α) τ (τ, t) τ=t =, λ (α) τ (τ, t) = and λ(τ, t) τ=t =. (4.5) The generalized Lagrange multiplier can be identified from Eq. (4.5) as λ(τ, t) = (τ t)α Γ( + α). (4.6) Substituting the above obtained Lagrange multiplier into Eq. (4.3) yields the iteration formulation as follows u n+ (x, t) = u n (x, t) + x (τ t) α Γ( + α) Γ( + α) ( D α τ u n (x, τ) + (u n(x, τ)) xx (u n(x, τ)) xxxx ) (dτ) α, n =,,.(4.7) Start with initial approximation: u (x, t) = u(x, ) + tα Γ(+α) Dα t u(x, ) = 4 3 sinh ( x 4)

5 5 ¹ 5 3 sinh( x ) t α Γ(+α) u (x, t) = u (x, t) + and we can derive x Γ( + α) (τ t) α = u (x, t) + Γ( + α) Γ( + α) = 3 cosh( x )( + t α ) 4 Γ( + α) u (x, t) = u (x, t) + = u (x, t) + Γ( + α) Γ( + α) (τ t) α ( D α Γ( + α) τ u,τ + (u (x, τ)) xx (u ) (x, τ)) xxxx (dτ) α ( sinh( x) τ α Γ( + α) 6 cosh( x) ) (dτ) α 3 sinh( x) ( τ α t 3α Γ( + α) + 8 Γ( + 3α) ) 3, (τ t) α ( D α Γ( + α) τ u,τ + (u (x, τ)) xx (u ) (x, τ)) xxxx (dτ) α (τ t) α Γ( + α) ( 48 sinh( x) t 3α Γ( + 3α) 4 cosh( x) t α ) (dτ) α Γ( + α) = 3 cosh( x )( + t α 4 Γ( + α) + t 4α ) 6 Γ( + 4α) 3 sinh( x) ( τ α Γ( + α) + τ 3α 8 Γ( + 3α) + t 5α ) 3 Γ( + 5α) 3. In the same manner further approximations of the iteration formula Eq. (4.7) can be obtained and more generally u n (x, t) = 3 cosh( x) n k= ( tα )k Γ( + kα) 3 sinh( x) n k= Importing the notation cosh α (x) = lim n n k= x k+ Γ(+(k+)α) [4], we can have a compact form ( tα )k+ Γ( + (k + )α) 3. (4.8) x k Γ(+kα) and sinh α(x) = lim n n k= u(x, t) = 3 cosh ( x (t coshα ) α ) 3 sinh( x (t sinhα ) α ) 3. (4.9) Substituting Eq. (4.9) into the Eq. (4.), we can verify easily that Eq. (4.9) is the exact solution of the fractional equation (4.). Specially, in the case of α = the solution (4.9) reduces into u(x, t) = 4 3 sinh ( 4 (x t) ), which is the single-soliton solution of the Boussinesq equation.

6 GU Jia-lei, et al: Handling the fractional Boussinesq-like equation 5 Fig. The surfaces are related with the solution of (4.) where α =. The figure only shows the trend of the approximate solutions on a large scale, which cannot illustrate the local behavior of fractional differential equations 5 Conclusion Variational iteration method is an efficient tool to solve nonlinear differential equations of integer order and it is successfully extended to solve fractional differential equation with modified Riemann-Liouville derivative. The key of VIM is how to identify the general Lagrange multiplier. After determining the general Lagrange multiplier, we can obtain a series of approximations to the exact solution step by step. È [] Decreusefond L, Ustunel A S. Stochastic analysis of the fractional Brownian motion [J]. Potential Anal., 999, : 77-4.

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