51. An Implicit Numerical Method for the Riemann–Liouville Distributed-Order Space Fractional Diffusion Equation
- Author
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Mengchen Zhang, Ming Shen, and Hui Chen
- Subjects
distributed-order operator ,diffusion equation ,fractional derivative ,implicit alternating direction method ,extrapolated technique ,Thermodynamics ,QC310.15-319 ,Mathematics ,QA1-939 ,Analysis ,QA299.6-433 - Abstract
This paper investigates a two-dimensional Riemann–Liouville distributed-order space fractional diffusion equation (RLDO-SFDE). However, many challenges exist in deriving analytical solutions for fractional dynamic systems. Efficient and reliable methods need to be explored for solving the RLDO-SFDE numerically. We develop an alternating direction implicit scheme and prove that the numerical method is unconditionally stable and convergent with an accuracy of O(σ2+ρ2+τ+hx+hy). After employing an extrapolated technique, the convergence order is improved to second order in time and space. Furthermore, a fast algorithm is constructed to reduce computational costs. Two numerical examples are presented to verify the effectiveness of the numerical methods. This study may provide more possibilities for simulating diffusion complexities by fractional calculus.
- Published
- 2023
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