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

Cheng, Rongjun; Ge, Hongxia; Wu, Yong

doi:https://doi.org/10.1155/2016/4893724

Mathematical Problems in Engineering

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Research Article | Open Access

Volume 2016 | Article ID 4893724 | https://doi.org/10.1155/2016/4893724

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

Rongjun Cheng,¹Hongxia Ge,¹and Yong Wu²

Academic Editor: Efstratios Tzirtzilakis

Received17 Dec 2015

Accepted22 Feb 2016

Published17 Mar 2016

Abstract

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.

1. 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 [13–15] 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 [24–26]. 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].

In the past few decades, meshless methods have already been a hot research topic in computational mechanics. Meshless methods also become important and powerful tools to research and analyze kinds of PDE. Researchers have presented many kinds of meshless methods, including the diffuse element method (DEM) [31], the smoothed particle hydrodynamics (SPH) method [32], the reproducing kernel particle method (RKPM) [33], the element-free Galerkin (EFG) method [34–37], the meshless local Petrov-Galerkin method (MLPG) [38–40], the hp-meshless cloud method [41], boundary element-free method (BEFM) [42, 43], the complex variable meshless method [44, 45], radius basis functions (RBF) [46], and LBIE method [47]. Now, meshless methods are widely used in various fields.

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 [50–53]. The IEFG method is a combination of IMLS approximation and EFG method [54–57]. 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 space-fractional equation with left and right Riemann-Liouville fractional derivatives. The IMLS-Ritz method for the two-sided 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.

2. IMLS Shape Functions

The local approximation is defined in the IMLS approximation [49] as follows:where is the number of bases, are monomial basis functions, and are corresponding coefficients. We can define the following quadratic form:where are compact weight functions and are the nodes. Equation (2) can be expressed in the matrix formwhereIn order to find the coefficients , we take the extremum of bywhich will get the following equation system:

If the functions meet with the following conditionsit will be called a weighted orthogonal function set with a weight function with points . The orthogonal function set can be obtained by using the Schmidt method, Equation (6) can be rewritten asThe coefficients can be easily founded as follows:that is,whereFrom (1), is expressed aswhere shape function is

Taking derivatives of (14), we will get derivatives of shape function

The cubic spline weight function is chosen as follows:wherewhere is called a scaling parameter and distance is chosen to make matrix which is no longer singular.

3. IMLS-Ritz Formulation for the Two-Sided Space-Fractional Wave Equation

Consider the following two-sided space-fractional wave equation:with initial conditionsand boundary conditionswhere the parameter describes the fractional order of spatial derivatives with . Function refers to a source term, and the coefficient functions and refer to transport related coefficients.

In order to establish the numerical approximation scheme, points , are considered, where . and are the boundary points; , , where is the time interval.

The left-handed and right Riemann-Liouville fractional derivatives of order are defined as [58]where is an integer such that .

The weighted integral form of (18a) is obtained as follows:

Define the energy functional as

Due to the boundary condition, the modified energy functional becomes

By (13), we can derive the approximate function as follows:whereSubstituting (23) into (22), by applying the Ritz minimization procedure to , we will derive

The results can be expressed aswhere

By the shifted Grünwald formula, we can discretize the Riemann-Liouville operator [18]where is the set of nodes and Gauss points and are the normalized Grünwald weights. The corresponding coefficients can be easily calculated by iteration formula as follows:

Substituting (28a), (28b), and (29) into (26) and discrete time by center difference method, we obtainwhere

The numerical solution of the space-fractional wave equation is obtained by iterative calculation.

4. 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 conditionswith boundary conditionswhere the source function isThe analytical solution isUsing IMLS-Ritz method to solve the equation with penalty factor , time step length , space step length , and . Table 1 shows numerical results obtained by the IMLS-Ritz method. The maximum error at time between the exact solution and the numerical solution and finite difference method [27] at different values of and is shown in Table 1. In Figure 1, the numerical and analytical solution are plotted at time ,, and , 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 conditionswith boundary conditionswhere the source function isThe exact solution isThe IMLS-Ritz method is applied to solve the above equation with penalty factor and time step length and . Table 2 shows numerical results obtained by the IMLS-Ritz method. The maximum error at time between the exact solution and the numerical solution and finite difference method [27] at different values of and is shown in Table 2. In Figure 4, the numerical and analytical solution are plotted at time ,, and , 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:with initial conditionswith boundary conditionsParticularly, if the fractional order of spatial derivatives , (42) will be a standard wave equation, and the exact solution to this problem in case is

Figures 7 and 8 show the numerical results with IMLS-Ritz method with and at different times and , respectively.

From these figures, it can be seen that the IMLS-Ritz method is very effective and accurate.

5. 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.

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

This project is supported by the Natural Science Foundation of Ningbo City (Grant nos. 2014A610028, 2015A610167, 2015A610168, 2015A610299, and 2015A610166) and the Natural Science Foundation of Zhejiang Province of China (Grant nos. LY15A020007, LQ13A010010) and the K. C. Wong Magna Fund in Ningbo University, China.

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