| Two Weighted Average Finite Difference Schemes for Variable-Order Fractional Mixed Diffusion and Diffusion-Wave Equation |
| Paper ID : 1031-ISCH |
| Authors |
|
Nada Henidy * Department of Mathematics, Obour Higher Institute of Management, Computers
and MIS, Cairo, Egypt. |
| Abstract |
| Recently, researchers have turned their attention toward variable-order derivatives, in which the derivative order depends on space and/or time. This extension offers a more accurate representation of systems exhibiting spatially or temporally varying memory effects [5]-[12]. Variable-order of PDEs (VOPDEs) are widely applied in biology, engineering, and finance. They are particularly useful for modeling complex phenomena such as sub-diffusion equations with the coefficients that are functions of spatial variables [5] and non-autonomous time-fractional diffusion equations involving a variable order depending on space [6]. Numerical methods for VOPDEs have seen significant advancements. For instance, Wang et al. [7] developed a discretization scheme for variable-order diffusion-wave equations, where the derivative order depends on space and/or time. Also, they combined Grünwald-Letnikov operators with Bernoulli polynomials to achieve accurate second-order convergence. Shen et al. [10] proposed a stable characteristic finite difference method for variable-order fractional advection-diffusion equations with nonlinear source terms, demonstrating improved computational performance. Moreover, the mixed diffusion and diffusion-wave equation with time-fractional derivatives has been extensively studied [3],[4]. This equation incorporates both fractional diffusion and diffusion-wave terms, where the fractional order lies within the ranges (0,1) and (1,2), respectively. To efficiently solve this equation, Sun et al. [3] developed a finite difference method based on the L1 approach. We analyze and compare two numerical methods: the weighted average standard finite difference method (WASFDM) and the weighted average nonstandard finite difference method (WANSFDM). The WASFDM employs a weighted factor [1, 2], while WANSFDM is proposed to enhance solution accuracy. |
| 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) |