A second‐order finite difference method for fractional diffusion equation with Dirichlet and fractional boundary conditions

Abstract

KEYWORDSCrank-Nicholson method, fractional boundary conditions, fractional diffusion equation, Riemann-Liouville fractional derivative, stability and convergence INTRODUCTIONFractional differential equations are generalizations of integer-order differential equations, which have been used in many models in physics, hydrology, geology, finance, and so on [1-6]. One of its most important applications is to describe, using fractional diffusion equations, the transport dynamics in various complex systems, where Gaussian statistics are no longer followed and the Fick's second law fails to describe the related transport behaviors [7]. A considerable number of numerical methods have been developed for space fractional diffusion equations [8][9][10][11][12][13][14][15]. Tadjeran et al. [8] used an approach based on the classical Crank-Nicholson method combined with spatial extrapolation to obtain temporally and spatially second-order accurate numerical estimates. In [9], combining the alternating directions implicit (ADI) approach with a Numer Methods Partial Differential Eq. 2019;35:1383-1395. wileyonlinelibrary.com/journal/num

Tempered fractional LES modeling

“…Fractional calculus introduces well-established mathematical tools for an accurate description of anomalous phenomena, ubiquitous in a wide range of applications from bio-tissues (Ionescu et al 2017;Naghibolhosseini & Long 2018) and material science (Meral, Royston & Magin 2010;Suzuki & Zayernouri 2021;Suzuki et al 2021a) to vibration (Suzuki et al 2021b), porous media (Xie & Fang 2019;Zaky, Hendy & Macías-Díaz 2020;Samiee et al 2020b). As an alternative approach to standard methods, they leverage their inherent potentials in representing long-range interactions, self-similar structures, sharp peaks and memory effects in a variety of applications (see Kharazmi & Zayernouri 2019;Burkovska, Glusa & D'Elia 2020).…”

Finite difference scheme for time‐space fractional diffusion equation with fractional boundary conditions

“…It is known that the weighted and shifted Grünwald-Letnikov operator can be used to approximate the Riemann-Liouville fractional derivative to the second-order accuracy. However, when the operator is used to approximate the space fractional derivative in the fractional boundary condition, because of the shift of the operator, there will appear points u(x ′ , t), where x ′ > L. In previous studies, [20][21][22][23] there are not similar case, as the space fractional derivative in the boundary condition are approximated by the standard Grünwald-Letnikov operator to first-order accuracy. Therefore, the difficulties in the paper are how to deal with u(x ′ , t) and the corresponding proof of stabilty and convergence.…”

“…Liu et al, developed an implicit finite difference method for fractional advection‐dispersion equations with fractional boundary conditions. Explicit and implicit Euler methods were generalized to two‐sided fractional diffusion in Kelly et al, Fang et al, developed a second‐order method to approximate the fractional diffusion equation with fractional boundary conditions, based on the Crank‐Nicolson method combined with spatial extrapolation. To our knowledge, numerical method for time‐space fractional diffusion equation with fractional boundary conditions is still limited.…”