A higher order numerical scheme for generalized fractional diffusion equations

• Published in: International journal for numerical methods in fluids Vol. 92; no. 12; pp. 1866 - 1889
• Main Authors: Ding, Qinxu, Wong, Patricia J. Y.
• Format: Journal Article
• Language: English
• Published: Bognor Regis Wiley Subscription Services, Inc 01-12-2020
• Subjects: Approximation; Convergence; Diffusion; diffusion problem; Energy methods; generalized fractional derivative; generalized weighted and shifted Grünwald‐Letnikov difference method; numerical solution; Stability; Stability analysis; Weighting functions
• ISSN: 0271-2091; 1097-0363
• DOI: 10.1002/fld.4852

In this article, we develop a higher order approximation for the generalized fractional derivative that includes a scale function z(t) and a weight function w(t). This is then used to solve a generalized fractional diffusion problem numerically. The stability and convergence analysis of the numerical scheme are conducted by the energy method. It is proven that the temporal convergence order is 3 and this is the best result to date. Finally, we present four examples to confirm the theoretical results.
We develop a higher order approximation for the generalized fractional derivative that features a scale function z(t) and a weight function w(t). This is then used to solve a generalized fractional diffusion problem numerically. We prove that the numerical scheme is stable and the temporal convergence order is 3, which is the best result to date.