Asymptotic Expansion of the Solutions to Time-Space Fractional Kuramoto-Sivashinsky Equations

This paper is devoted to finding the asymptotic expansion of solutions to fractional partial differential equations with initial conditions. A new method, the residual power series method, is proposed for time-space fractional partial differential equations, where the fractional integral and derivative are described in the sense of Riemann-Liouville integral and Caputo derivative. We apply the method to the linear and nonlinear time-space fractional Kuramoto-Sivashinsky equation with initial value and obtain asymptotic expansion of the solutions, which demonstrates the accuracy and efficiency of the method.


Introduction
The Kuramoto-Sivashinsky (KS) equation in one space dimension,    (, ) +  4    (, ) +  2   (, ) +  (, )    (, ) = 0, has attracted a great deal of interest as a model for complex spatiotemporal dynamics in spatially extended systems and as a paradigm for finite-dimensional dynamics in a partial differential equation. 2   term in (1) is responsible for an instability at large scales; dissipative term  4    provides damping at small scales; and the nonlinear term    stabilizes by transferring energy between large and small scales.The KS equation dates back to the mid-1970s and was first introduced by Kuramoto [1] in the study of phase turbulence in the Belousov-Zhabotinsky reaction-diffusion systems.An extension of this equation to two or more spatial dimensions was given by Sivashinsky [2,3] in modelling small thermal diffusive instabilities in laminar flame fronts and in small perturbations from a reference Poiseuille flow of a flame layer on an inclined plane.In one space dimension it is also used as model for the problem of Bénard convection in an elongated box, and it may be used to describe long waves on the interface between two viscous fluids and unstable drift wave in plasmas.As a dynamical system, KS equation is known for its chaotic solutions and complicated behavior due to the terms that appear.Because of this fact, KS equation was studied extensively as a paradigm of finite dynamics in a partial differential equation.Its multimodal, oscillatory, and chaotic solutions have been investigated [4][5][6][7][8]; its nonintegrability was established via Painlevé analysis [9] and due to its bifurcation behavior a connection to low finite-dimensional dynamical systems is established [10,11].The KS equation is nonintegrable; therefore the exact solution of this equation is not obtainable and only numerical schemes have been proposed [12,13].
Partial differential equations (PDEs) which arise in realworld physical problems are often too complicated to be solved exactly and even if an exact solution is obtainable, the required calculations may be too complicated to be practical or difficult to interpret the outcome.Very recently, some practical approximate analytical solutions are proposed to 2 Advances in Mathematical Physics solve KS equation, such as Chebyshev spectral collocation scheme [14], lattice Boltzmann technique [15], local discontinuous Galerkin method [16], tanh function method [17], variational iteration method [18], perturbation methods [19], classical and nonclassical symmetries method for the KS equation dispersive effects [20], Riccati expansion method [21], and Lie symmetry method [22], and see also [23][24][25][26][27].In the last few decades, fractional-order models are found to be more adequate than integer-order models for some realworld problems.Fractional derivatives provide an excellent tool for the description of memory and hereditary properties of various materials and processes.This is the main advantage of fractional differential equations in comparison with classical integer-order models.Fractional differential equations arise in many engineering and scientific disciplines as the mathematical modelling of systems and processes in the fields of physics, chemistry, aerodynamics, electrodynamics of complex medium, polymer rheology, and so forth involves derivatives of fractional order.In particular, for the construction of the approximate solutions of the fractional PDEs, various methods are proposed: finite difference method [28], the finite element method [29], the differential transformation method [30], the fractional subequation method [31], the fractional complex transform method [32], the modified simple equation method [33], the variational iteration method [34], the Lagrange characteristic method [35], the iteration method [36], and so on.For the time-fractional KS equation, in [37] the authors constructed the analytical exact solutions via fractional complex transform, and they obtained new types of exact analytical solutions.In [38], Rezazadeh and Ziabary found travelling wave solutions by the general time-space KS equation by a subequation method.
In this work, we apply residual power series (RPS) method to construct the asymptotic expansion of the solution to the more general linear KS equation with initial value where 0 < ,  ≤ 1, 3/4 <  ≤ 1, 1/2 <  ≤ 1, , , and  are any arbitrary constants, and (, ) ∈ R × R. The general response expression contains different parameters describing the order of the fractional derivative that can be varied to obtain various responses.The fractional power series solutions can be obtained by the RPS method.Particularly, if we take special parameters  = 1,  = 1,  = 1,  = 1,  = 1,  = 1, and  = 1 (here the equation is integer-order) and special initial condition  0 () = , the exact solution of linear KS equation is and the exact solution of nonlinear KS equation also can be obtained: Here the solution of nonlinear KS equation we obtained is different from Porshokouhi and Ghanbari's work in [39] for the integer-order KS equation.
of ( 2) with the same special parameters  = 1,  = 1/4,  = 1,  = 1,  = 1,  = 1, and  = 1.Their skills mainly depend on the variational iteration method and obtain the different numerical examples with different parameters.To the best of information of the authors, no previous research work has been done using proposed technique for solving time-space fractional KS equation.Our method can be applied to the time-space linear and nonlinear fractional KS equations.The main advantage of the RPS method is that it can be applied directly for all types of differential equation, because it depends on the recursive differentiation of time-fractional derivative and uses given initial conditions to calculate coefficients of the multiple fractional power series solution with minimal calculations.Another important advantage is that this method does not require linearization, perturbation, or discretization of the variables; it is not affected by computational round-off errors and does not require large computer memory and extensive time.The rest of this paper is organised as follows.In Section 2, some necessary concepts on the theory of fractional calculus are presented.The main steps of the PRS method are proposed in Section 3. Section 4 is the application of RPS method to construct analytical solution of linear and nonlinear timespace fractional KS equation with initial value.The paper is concluded with some general remarks in Section 5.

Some Concepts on the Theory of Fractional Calculus
There are several definitions of the fractional integral and fractional derivative, which are not necessarily equivalent to each other (see [40][41][42]).Riemann-Liouville integral and Caputo derivative are the two most used forms which have been introduced in [43][44][45].In this section, we give some notions we need in this paper.
Remark 5 (see [41,46]).The Caputo space-fractional derivative operator of order  of function (, ) with respect to  at  0 is defined as where  ∈ R and  > 0.
Definition 6 (see [41,46]).A power series representation of the form where  − 1 <  ≤  and  ⩾  0 , is called a fractional power series (FPS) about  0 , where  is a variable and   are constants called the coefficients of the series.
Theorem 7. Suppose that  has a FPS representation at  0 of the form and  is the radius of convergence of the FPS.If () ∈ [ 0 ,  0 + ) and  0    () ∈ ( 0 ,  0 + ) for  = 0, 1, 2, . .., then the coefficients   will take the form of where This result is similar to [46, Theorem 2.2] and [47, Theorem 3.4].It is convenient to give the details for the following applications; thus we write the process of the proof in the form of function with one variable.
Proof.First of all, notice that if we put  =  0 into (16), it yields Applying the operator  0    one time on (16) leads to Again, by applying the operator  0    two times on (16), one can obtain By now, the pattern is clearly found, if we continue applying recursively the operator  0    -times on ( 16); then it is easy to discover the following form for   : ,  = 0, 1, 2, . . .; that is, it has the same form as (17), which completes the proof.

Algorithm of RPS Method
Let us consider the higher-order time-space fractional differential equation with initial values as follows: where  ∈ R,  ∈ R, and  Assume that (, ) is analytical about ; the solution of the system can be written in the form of where   (, ) are terms of approximations and are given as where  is the radius of convergence of above series.Obviously, when  = 0, 1, 2, . . .,  − 1, using the terms    (, ) which satisfy the initial condition, we can get ( So we have the initial guess approximation of (, ) in the following form: On the other aspect as well, if we choose  initial (, ) as initial guess approximation   (, ) for  = ,  + 1,  + 2, . .., the approximate solutions of (, ) of ( 23 Before applying RPS method for solving (23), we first give some notations: Res (, ) fl     (, ) +  (, ) .

Application of RPS Method to Time-Space KS Equation
In this section, we apply the RPS method to the linear and nonlinear time-space fractional KS equation with the initial conditions.The fractional power series solutions can be obtained by the recursive equation ( 35) with time-fractional derivative, while it will use the given initial conditions.And the fractional power series solutions we consider in the following examples are all in the convergence radius of the series.

Nonlinear
If (, ) is analytical about  of (49), then (, ) can be written as the fractional power series: Here, the initial value satisfies According to the initial condition, it is obvious that  0 () =  0 ().

Concluding Remarks
In this paper, we have used a new method: the residual power series method for the general fractional differential equations.The asymptotic expansion of the solutions can be obtained successfully with respect to initial conditions which are infinitely differentiable by the RPS method.We apply RPS method to linear and nonlinear Kuramoto-Sivashinsky equation with infinitely differential initial conditions and obtain the asymptotic expansion of the solutions.Particularly, if taking special parameters and special initial value, the analytical solutions are obtained.These applications show that this method is efficient and does not require linearization or perturbation; it is not affected by computational roundoff errors and does not require large computer memory and extensive time.