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.
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Dehestani, H. and Ordokhani, Y. (2025). A highly accurate wavelet approach for multi-term variable-order fractional multi-dimensional differential equations. Computational Methods for Differential Equations, 13(3), 850-869. doi: 10.22034/cmde.2024.62926.2793
MLA
Dehestani, H. , and Ordokhani, Y. . "A highly accurate wavelet approach for multi-term variable-order fractional multi-dimensional differential equations", Computational Methods for Differential Equations, 13, 3, 2025, 850-869. doi: 10.22034/cmde.2024.62926.2793
HARVARD
Dehestani, H., Ordokhani, Y. (2025). 'A highly accurate wavelet approach for multi-term variable-order fractional multi-dimensional differential equations', Computational Methods for Differential Equations, 13(3), pp. 850-869. doi: 10.22034/cmde.2024.62926.2793
CHICAGO
H. Dehestani and Y. Ordokhani, "A highly accurate wavelet approach for multi-term variable-order fractional multi-dimensional differential equations," Computational Methods for Differential Equations, 13 3 (2025): 850-869, doi: 10.22034/cmde.2024.62926.2793
VANCOUVER
Dehestani, H., Ordokhani, Y. A highly accurate wavelet approach for multi-term variable-order fractional multi-dimensional differential equations. Computational Methods for Differential Equations, 2025; 13(3): 850-869. doi: 10.22034/cmde.2024.62926.2793
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