A highly accurate wavelet approach for multi-term variable-order fractional multi-dimensional differential equations

In this work, the multi-term variable-order fractional multi-dimensional differential equations are studied based on Gegenbauer wavelet functions. The main aim of this paper is to develop the spectral method with the help of modified operational matrices, which are directly effective in the numerical process. Therefore, we discuss the novel method of obtaining the modified operational matrices (MOMs) of integration and variable-order (VO) fractional derivative. Then, the overall algorithm for solving multi-term VO-fractional differential equations and partial differential equations is introduced. We also discuss error analysis in detail. At last, we implement the numerical scheme in several examples that involve the damped mechanical oscillator equation, the VO-fractional mobile-immobile advection-dispersion equation, and the VO-fractional nonlinear Galilei invariant advection-diffusion equation. Also, to confirm the theoretical results and demonstrate the accuracy and efficiency of the method, we compare our numerical results with analytical solutions and other existing methods.

کلیدواژه ها:

Gegenbauer wavelet functions ، Modified operational matrix ، Transformation matrix ، Variable-order fractional derivative ، partial differential equations