The Numerical Analysis of Two-Sided Space-Fractional Wave Equation with Improved Moving Least-Square Ritz Method

A numerical analysis of the space-fractional wave equation is carried out by the improved moving least-square Ritz (IMLS-Ritz) method. The trial functions for the space-fractional wave equation are constructed by the IMLS approximation. By the Galerkin weak form, the energy functional is formulated. Employing the Ritz minimization procedure, the final algebraic equations system is obtained. In this numerical analysis, the applicability and efficiency of the IMLS-Ritz method are examined by some example problems. Comparing the numerical results with the analytical solutions, the stability and accuracy of the IMLS-Ritz method are also presented.


Introduction
Due to extensive use in the fields of dynamics [1], fluid mechanics [2], viscoelasticity [3], materials [4], hydrology [5], biology [6,7], porous media [8], physics [9,10], engineering [11,12], and so on, fractional partial differential equations have become a hot research topic.Consequently, scholars pay much attention to the analytical solutions of fractional partial differential equation (PDE) of physical interest.Unfortunately, a fractional PDE has no exact solution in many cases owing to complex series or special functions.So it is extremely important and necessary to resort to numerical solutions.
The fractional diffusion-wave equation has been an interesting topic to invest in during the past decades.There is already some important progress for the fractional diffusion equation or advection-diffusion equations.Deng [13][14][15] presented the numerical method for fractional diffusion equations, Liu et al. [16] used the difference method for space-time fractional equation and presented the stability and convergence, Meerschaert and Tadjeran [17] applied the finite difference approximation for space-fractional equations, Meng [18] put forward a new approach for solving fractional partial differential equations, Sousa [19] developed numerical approximations for fractional diffusion equations via splines, Zhou and Wu [20] proposed the finite element multigrid method for the boundary value problem of fractional advection dispersion equation.Nigmatullin [21] presented the fractional diffusion equation to describe the diffusion in porous media.Mainardi [22,23] has shown that the FWE describes the propagation of mechanical diffusive waves in viscoelastic media.Much study has been done for the time fractional PDE in [24][25][26].In fact, most of the works focus on the time fractional PDE.In another latest paper, Sweilam et al. considered a 1D fractional wave equation in [27], Deng et al. used the alternating direction implicit algorithm for the space-fractional equation in [28], Jia and Wang used the fast finite difference methods for space-fractional PDE with fractional derivative boundary conditions in [29], and Guan and Gunzburger applied the finite element method (FEM) for the space-time fractional PDE in [30].
The MLS approximation originated from data fitting [48].In recent years, the MLS technique is often used for analysis of solid mechanics problems in the meshless method or EFG method [48].Unfortunately, the final algebraic equations system obtained by EFG method may be ill-conditioned.Thus, the ill-conditioned algebraic equations system should be considered in the MLS approximation.Unless we solve the equation, it is hard to confirm whether the algebraic equations system is ill-conditioned or not.Therefore, it could lead to poor or erroneous numerical results.In order to overcome this problem, Cheng and Chen proposed the IMLS approximation [49].In the IMLS approximation, the orthogonal function system is chosen as the basis function, and the resulting algebraic equation system is not ill-conditioned any more.Based on the IMLS approximation, the boundary element-free method is presented for elasticity, fracture, elastodynamics, and potential problems [50][51][52][53].The IEFG method is a combination of IMLS approximation and EFG method [54][55][56][57].The IEFG method needs fewer nodes than the conventional EFG method.Hence, the IEFG method is bounded to increase the computational speed and has higher accuracy than the EFG method.
As far as is known, the space-fractional PDE has never been analyzed and researched by the IMLS-Ritz method.We try to consider the one-dimensional two-sided spacefractional equation with left and right Riemann-Liouville fractional derivatives.The IMLS-Ritz method for the twosided space-fractional wave equation is put forward.In this paper, the IMLS approximation is used to approximate displacement field, the penalty method is applied to impose the boundary conditions, and Ritz minimization procedure is used to obtain the final algebraic equation system.In order to verify the validity and stability of the proposed method, numerical examples are presented compared with existing results available in extant literature.

IMLS Shape Functions
The local approximation is defined in the IMLS approximation [49] as follows: where  is the number of bases,   (x) are monomial basis functions, and   (x) are corresponding coefficients.We can define the following quadratic form: where (x − x  ) are compact weight functions and x  are the nodes.Equation ( 2) can be expressed in the matrix form where In order to find the coefficients a(x), we take the extremum of  by which will get the following equation system: If the functions  1 (x),  2 (x), . . .,   (x) meet with the following conditions it will be called a weighted orthogonal function set with a weight function {  } with points {x  }.The orthogonal function set p = (  ) can be obtained by using the Schmidt method, Mathematical Problems in Engineering 3 Equation ( 6) can be rewritten as The coefficients   (x) can be easily founded as follows: that is, where From (1),  ℎ (x) is expressed as where shape function Φ(x) is Taking derivatives of ( 14), we will get derivatives of shape function The cubic spline weight function is chosen as follows: where where  max is called a scaling parameter and distance   is chosen to make matrix () which is no longer singular.

IMLS-Ritz Formulation for the Two-Sided Space-Fractional Wave Equation
Consider the following two-sided space-fractional wave equation: with initial conditions  (, 0) =  0 () , and boundary conditions where the parameter  describes the fractional order of spatial derivatives with 1 <  ≤ 2. Function (, ) refers to a source term, and the coefficient functions  + () > 0 and  − () > 0 refer to transport related coefficients.
The left-handed and right Riemann-Liouville fractional derivatives of order  are defined as [58] where  is an integer such that  − 1 <  ≤ .
The weighted integral form of (18a) is obtained as follows:

Numerical Results
In order to verify the validity and correctness of the proposed IMLS-Ritz method for the space-fractional wave equation, examples are studied and the numerical results are presented.Note that, in all examples considered, the cubic spline function is chosen as weight function and the linear bases are chosen in this paper.
Example 1 (left-handed space-fractional wave equation).Consider the following left-handed space-fractional wave equation:  with initial conditions with boundary conditions where the source function is The analytical solution is Using IMLS-Ritz method to solve the equation with penalty factor  1 = 10 7 , time step length Δ = 0.001, space step length Δ = 0.0125, and  max = 3.8.Table 1 shows numerical results obtained by the IMLS-Ritz method.The maximum error at time  = 1 between the exact solution and the numerical solution and finite difference method [27] at different values of Δ and Δ is shown in Table 1.In Figure 1, the numerical and analytical solution are plotted at time  = 0.5,1, and 1.5, respectively.The surface of the numerical and analytical solution is plotted in Figures 2 and 3, respectively.Numerical results show that the IMLS-Ritz method is very effective and accurate.Example 2 (two-sided space-fractional).Consider the following left-handed and right-handed space-fractional wave equation: with initial conditions with boundary conditions  where the source function is The exact solution is The IMLS-Ritz method is applied to solve the above equation with penalty factor  1 = 10 7 and time step length Δ = 0.002 and  max = 3.8.Table 2 shows numerical results obtained by the IMLS-Ritz method.The maximum error at time  = 2 between the exact solution and the numerical solution and finite difference method [27] at different values of Δ and Δ is shown in Table 2.In Figure 4, the numerical and analytical solution are plotted at time  = 0.5,1, and 2, respectively.The surface of the numerical and analytical solution is plotted in Figures 5 and 6, respectively.Numerical results show that the IMLS-Ritz method is very effective and accurate.Example 3 (two-sided space-fractional).Consider the following left-handed and right-handed space-fractional wave equation: Particularly, if the fractional order of spatial derivatives  = 2, (42) will be a standard wave equation, and the exact solution to this problem in case  = 2 is  (, ) = sin () cos (2) .
(45)  From these figures, it can be seen that the IMLS-Ritz method is very effective and accurate.

Conclusions
The meshless method for the two-sided space-fractional wave equation is put forward in this paper.In the present method, the IMLS approximation is employed to construct the shape functions.In the IMLS, the orthogonal function system with a weight function is chosen as the basis function.Through employing the Ritz minimization procedure to the energy expressions, the final algebraic equations system is obtained.The system obtained by IMLS technique will be not ill-conditioned any more, and the solution can be easily obtained without matrix inversion.Because of the simplicity of numerical implementation, the proposed IMLS-Ritz method will substitute for the difference method and the finite element method for solving space-fractional wave equation and other fractional partial differential equations.

Figures 7 and 8
Figures 7 and 8 show the numerical results with IMLS-Ritz method with  = 2 and  = 1.8 at different times  = 1 and  = 1.5, respectively.From these figures, it can be seen that the IMLS-Ritz method is very effective and accurate.