The meshless method for two-dimensional space-time fractional dispersion equation based on reproducing kernel particle method

Abstract

The two-dimensional space fractional dispersion equation (SFDE) is obtained from the standard dispersion equation by replacing the two second-order space derivatives by the Riemann–Liouville fractional derivatives. A numerical analysis of the two-dimensional SFDE is presented based on the reproducing kernel particle method (RKPM). The final algebraic equation system is obtained by employing Galerkin weak form and functional minimization procedure. The Riemann–Liouville operator is discretized by the shifted Grünwald formula. The fully-discrete approximation schemes for SFDE are established using center difference method and RKPM and the shifted Grünwald formula. Numerical simulations for SFDE with known exact solution were presented in the format of the tables and graphs. The presented results demonstrate the validity, efficiency and accuracy of the proposed techniques. Furthermore, the error estimate of RKPM for SFDE has been analyzed, which shows that this method has reasonable convergence rates in spatial and temporal discretizations.