| Two Weighted Average Finite Difference Schemes for Variable-Order Fractional Mixed Diffusion and Diffusion-Wave Equation |
| Paper ID : 1040-ISCH |
| Authors |
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Nada Henidy *1, Nasser Hassen Sweilam2, Salma Asaad Shatta3, Adel Darwish4 1T.A at Department of Mathematics, Obour Higher Institute of Management, Computers and MIS, Cairo, Egypt 2Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt 3Instructor at Department of Mathematics, Faculty of Science, Helwan University, Cairo, Egypt 4Department of Mathematics, Faculty of Science, Helwan University, Cairo, Egypt |
| Abstract |
| Recently, increasing attention has been given to variable-order partial differential equations (VOPDEs), in which the order of the derivative depends on space and/or time. This framework provides a more accurate description of systems exhibiting spatially or temporally varying memory effects, with applications spanning biology, engineering, and finance. In particular, VOPDEs are effective for modeling complex processes such as sub-diffusion with spatially dependent coefficients and non-autonomous time-fractional diffusion equations. Significant progress has been made in developing numerical methods for VOPDEs. For instance, Wang et al. proposed a discretization scheme for variable-order diffusion-wave equations by combining Grünwald–Letnikov operators with Bernoulli polynomials, achieving second-order convergence. Shen et al. introduced a stable characteristic finite difference method for variable-order fractional advection–diffusion equations with nonlinear sources, while Sun et al. applied the L1 approach to efficiently solve mixed diffusion and diffusion-wave equations with time-fractional derivatives. In this work, we analyze and compare two approaches for solving VOPDEs: the weighted average standard finite difference method (WASFDM) and the weighted average nonstandard finite difference method (WANSFDM). While WASFDM employs a weighted factor, WANSFDM is designed to enhance numerical accuracy. Our comparative study highlights the performance, stability, and accuracy of these methods in capturing the dynamics of variable-order fractional systems. |
| Keywords |
| Caputo derivative; Mixed diffusion and diffusion-wave model problem; Nonstandard and standard weighted average finite difference methods; Stability analysis. |
| Status: Abstract Accepted (Poster Presentation) |