ON THE EVENTUAL PERIODICITY OF FRACTIONAL ORDER DISPERSIVE WAVE EQUATIONS USING RBFS AND TRANSFORM

In this research work, let’s focus on the construction of numerical scheme based on radial basis functions finite diffe­ rence (RBF­FD) method combined with the Laplace transform for the solution of fractional order dispersive wave equations. The numerical scheme is then applied to examine the eventual periodicity of the proposed model subject to the periodic boundary conditions. The implementation of proposed technique for high order fractional and integer type nonlinear partial differential equations (PDEs) is beneficial because this method is local in nature, therefore it yields and resulted in sparse differentiation matrices instead of full and dense matrices. Only small dimensions of linear systems of equations are to be solved for every center in the domain and hence this procedure is more reliable and efficient to solve large scale physical and engineering problems in complex domain. Laplace transform is utilized for obtaining the equivalent time­independent equation in Laplace space and also valuable to handle time­fractional derivatives in the Caputo sense. Application of Laplace transform avoids the time steeping procedure which commonly encounters the time instability issues. The solution to the transformed model is then obtained by computing the inversion of Laplace transform with an appropriate contour in a complex space, which is approximated by trapezoidal rule with high accuracy. Also since the Laplace transform operator is linear, it cannot be used to transform non­linear terms therefore let’s use a linearization approach and an appropriate iterative scheme. The proposed approach is tasted for some nonlinear fractional order KdV and Burgers equations. The capacity, high order accuracy and efficiency of our approach are demonstrated using examples and results .


Introduction
Significant advancement to both the theory and application of fractional calculus has been made in the last century. The most essential advantage of fractional calculus is its nonlocal nature and effectiveness in modeling anomalous diffusion that happens in complex system transport dynamics, such as fluid flow in viscoelastic material [1], porous materials [2,3], anomalous transport in biology [4], etc. Also fractional calculus is used to model problems in control theory, image processing, entropy theory and wave propagation phenomenon [5][6][7]. Due to large application in engineering, physics and mathematical sciences the analytical and numerical techniques for solving differential equations of fractional order increased dramatically in the recent year [8][9][10][11]. To learn more about fractional calculus, such as its existence, uniqueness of solution, applicability and solution methods, etc, [7,[12][13][14]. Mathematics Many natural phenomena have been successfully modeled using fractional differential equations (FDEs) [7,12,15]. Physical phenomena including traffic flow models with fractional derivatives [16], fractional order KdV equation [9], fractional advection dispersion equation [10], and shock waves in a viscous medium [17] can be represented using nonlinear differential equations.
The fractional derivative is a nonlocal operator, discretizing the temporal fractional derivative is a difficult task. To numerically solve FDEs in the complex domain and anticipate longtime ranges, an efficient numerical approach is required. Let's employ the Laplace transform to solve these problems because the temporal fractional derivatives are easily handled by the Laplace transform. Radial basis function generated finite difference (RBF-FD) method will be used to treat the spatial variable.
Mesh-free methods are becoming more popular, emerging, interesting and fascinating numerical techniques due to the ability to solve those physical and engineering problems with no meshing or minimum of meshing for which the traditionally used mesh-based methods are not suited like finite volumes, finite differences, finite elements, moving least square, point interpolation, element free Galerkin, reproducing kernel particle and boundary element free methods. RBFs methods appears to be really consists and most prominent techniques among the meshless methods family while looking at the interpolation of multidimensional scattered data and have received recently a tremendous and considerable attention in scientific community because of its capacity to achieve spectral accuracy, efficiency and high flexibility in solving complex PDEs, integral equations and fractional equations in comparison to other advanced approaches [18][19][20][21]. The most commonly used kernel in meshless techniques is the multiquadric (MQ) kernel suggested by [21][22][23][24]. [22,23] using the radial basis function for solution of a collocation strategy for PDEs.
Another specific qualitative characteristic disclosed on solutions to IBVPs of some evolutionary equations that have been established through investigations and are linked by their large time stunts the term eventual time periodicity was coined to describe this phenomenon. This enticing and appealing event take place by a piston or flap or paddletype wave maker put on one of the channel's ends in tests of research. When the wave generator oscillates at a predictable period T 0 > 0, at each location along the channel, the amplitude of the waves appears to become periodic, when a particular period of time has elapsed [25][26][27][28]. Various studies have previously addressed these important and fascinating eventual periodic phenomena such as Burgertype equations, generalized equations for KdV, BBM, and its dissipating counterparts [29][30][31][32][33][34][35][36].
In this work, let's offer an iterative strategy for obtaining approximate solutions of non linear time fractional KdV and Burgers equation using the Laplace transform in conjunction with RBFFD meshless method. Also the said scheme is implemented to investigate eventual periodicity to these equations.

Materials and methods 1. Preliminaries Definition 1.
The fractional order derivative in the Caputo sense [12,37] is defined by: where n n z -

Definition 2.
The Laplace transform of a given function w(t) where t ≥ 0 is defined as: if improper integral converges. (3)
is the initial condition and u x t g t , , are the boundary conditions with boundary operator .

3. Description of the method Description of RBF-Finite Differences method.
Let's deal with general time dependent PDE for mathematical formulation and define the RBFFD process in a gradual way. Take the problem of frame: , , associated with initial and boundary conditions: , , where u 0 and h are certain provided functions, while the spatial operators L,  representing the differential operators. Assume { } denotes N number of nodes used for approximation in the domain Ξ for the given problem. RBFFD is a meshfree method and essentially a generalization of conventional finite difference (FD) method. In classical FD approach the derivative of a function u is defined as a linear combination of the values of u at some closest surrounding values (stencil) nodes. The difference is that RBF-FD methods use radial basis function instead of polynomials use in classical FD method [24].
Global RBF differentiation matrix. Discretization of equations (8), (9) via global RBF method can be followed by approximating the unknown function u by the linear combination of radial kernel ϕ at the node x specified by: Mathematics such that: and C is the expansion coefficients vector. Now equation (10) in Lagrange form is stated as: Here K representing system interpolation matrix for the global RBF. Now the interpolant (kernelbased)  u in (12) gives good approximation of u. Consequently, any operator used on  u also would be an excellent estimation of relevant operator employed on u [21,24]. Applying linear differential operator L on above equation (12) gives: From (13) let's use the notation below for values: The global discretization (differentiation) matrix L of size N×N may thus be considered as: Since from (14), it is possible to see that the i th row of K L corresponds to LF(x i ) T , therefore let's observe from (15) that the i th row of L, serve as global differentiation matrix L one single row. Local RBF differentiation matrix. Let's now report derivation of local differentiation matrix and describe how to compute the local finite differences associated weights which give rise to local interpolant in a locally small neighborhood regarding point x i exactly. Let's consider the set of points Ξ = {x 1 , x 2 , …, x N }, where it is possible to make the derivative approximation, these points can be regarded as stencil centers. For a given i th evaluation node say x i , the size of nearest neighboring nodes in stencil N x i of x i is n. Specifying also the set of points Z = {z 1 , z 2 , …, z N } at which it is possible to analyze (sample) data. The points inside the stencil having size n are collected at Z i ⊂ Z. Now estimation of differential operator L on stencil with center node x i and collected at Z i is given by: Actually it assembles a stencil having center node x i hence to declare it as local differentiation matrix however it behaves globally since it operate whole entire data of that small stencil. All those L i matrices contains nonzeros entries in sparse (global) matrix L FD , however their position must still be determined further in that sparse matrix L FD . Now L i FD which representing the i th row of L FD and holds nonzero values from matrix L i (since it has one test node x i so it is row vector). As the points in Z i ⊂ Z are used in constructing L i (Hence columns of L FD connected along with those points which are nonzero columns th (18) Use this to describe the complete sparse matrix as: Ultimately the discretization for problem (8), (9) maybe written as: FD FD , B FD stand for the discretization of operator applied at the boundary and can accordingly be found as L FD . Evolving in time the ODE system (20), some solver ODE such as, ode113 ode23, ode45, and several others can be used from Matlab.
Time discretization via Laplace transform. Now discuss how to reduce the non-linear fractional order differential equation (5)-(7) to system of nonlinear algebraic equations. The resulting system is then solved with the help of RBFFD approach on a local level. The use of Laplace transform eliminates the time variable. Since the Laplace transform is a linear operator, it can't be directly applied to equations that are nonlinear, therefore let's design an iterative approach for linearizing the nonlinear equation so that the Laplace transform can be applied to each step of the iterative process. It is possible to of the problem (5)-(7) as given below: Applying Laplace transform to (21): and boundary conditions: Now introducing the differential operators  1 ,  2 and  3 : Thus the nonlinear iterative update process to approximate the solution u(x,s), using u 0 as the initial approximation is given by:  Mathematics where  u x s n ( , ) is the n th iterate of the solution. The iterations will continue when the change in the solution is less than or equal to some preassign value. The problem (25) is approximated using radial basis functions in FD setting. An inverse Laplace transform approach is used for getting the solution u(x,t) of the given model (21).
Using the parabolic path defined in (30), integral in (26) becomes: Applying the trapezoidal rule with equal step size the approximation to (32) is given by: where h is the step size.

1. Numerical Results of our proposed scheme
This section is devoted to demonstrate the accuracy and efficiency of our proposed numerical scheme called as radial basis function finite difference method combined with Laplace transform (RBFFD)LT by solving a number of nonlinear testing challenges. The main advantage of our method is avoidance of time stepping procedure which need a small time step for higher accuracy and stability, the second advantage is that our method has the capability to work in multidimensions in irregu lar domain as well. The accuracy of our method is tested in terms of L ∞ , L 2 , root mean square (RMS) and E(c, M) errors norms. Comparison with available results in literature for example in reference [38], Tables 1-4 and Fig. 1-9, shows that our method is more accurate and stable. Let's solve some benchmark problems like time fractional KdV, Burger and KdVBurger equations in one dimension as under.
Here let's consider the domain Ω = [0, 1]. The contour mentioned in (30) is used here and is shown in Fig. 4.

Mathematics
Problem 3: finally, look at the basic Burger equation [42] which is given by: The initial condition is: and are the boundary conditions. The exact solution is: exp sin exp c os , Here take Ω = [0,1]. The parabolic contour (30) is used here. The parameter values t = 0.001, ξ = 0.1, 0.2, 0.5, 1, and N = 41, and as given in [42] are used. Fig. 6 depicts the numerical solutions found using our proposed method, the results of the numerical technique are identical to those in [42].

2. Eventual Periodicity of the proposed model equations
Now let's present the results of our method investigating the eventual periodicity of the given model (5)-(7) for KdV, Burger and KdVBurger equations studied by [30] in graphical form along with appropriate boundary data g(t) (periodic function of period T 0 > 0). The initial data u 0 is not necessarily necessary in eventual periodicity so take it zero. For each problem the amplitudes u(x,t) produced in six graphs at particular points. N indicates complete domain points, while N x denotes points in respective subdomains. The X and Y axes are representative in these graphs of time t and amplitude u respectively. The last graph shows the amplitude remains zero in every problem.
Eventual periodicity of fractional KdV equation: let's compute eventual periodicity of model equation (5)   The plots below clearly confirm the subsequent periodic activity of the solution in the specified domain at these particular positions. The pattern of eventual periodicity has not changed, but the amplitudes have been drastically reduced.

Conclusions
In this research work, a hybrid numerical scheme is developed by combining the Laplace transform technique with RBFFD mesh free method for approximating the solution of some nonlinear time fractional dispersive wave equations (like KdV, Burger and KdVBurger model) and also the periodic behavior of solution called as eventual periodicity. This method is integrated with Laplace transform approach for time integration and for fractional derivative in Caputo sense. The main advantage of Laplace transform technique is that it is free of issues relating to stability due to time steeping scheme. The spatial operators in multidimensions are approximated by RBF in the finite difference (FD) setting which generates small size differentiation matrices in local sub domains and these are assembled as a single sparse matrix in the global domain. So large amount of data can be manipulated very easily and accurately. The construction of our approach is simpler and easier to solve any nonlinear higher order time fractional PDEs as compared to other numerical methods available in the literature. The simulation has been demonstrated through error norms recorded in tables and graphs, which shows the accuracy and robustness of our method.