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.
Highlights
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
A fractional partial differential equation (PDE) has no exact solution in many cases owing to complex series or special functions
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
Summary
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. The final algebraic equations system obtained by EFG method may be ill-conditioned. The ill-conditioned algebraic equations system should be considered in the MLS approximation. 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. As far as is known, the space-fractional PDE has never been analyzed and researched by the IMLS-Ritz method. The IMLS-Ritz method for the twosided space-fractional wave equation is put forward. 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
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