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A C α finite difference method for the Caputo time‐fractional diffusion equation
Author(s) -
Davis Wesley,
Noren Richard,
Shi Ke
Publication year - 2021
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.22686
Subject(s) - mathematics , laplace transform , convergence (economics) , variable (mathematics) , order (exchange) , diffusion equation , fractional calculus , mathematical analysis , rate of convergence , differential equation , integro differential equation , diffusion , first order partial differential equation , physics , quantum mechanics , channel (broadcasting) , economy , finance , service (business) , economics , economic growth , engineering , electrical engineering
We begin with a treatment of the Caputo time‐fractional diffusion equation, by using the Laplace transform, to obtain a Volterra integro‐differential equation. We derive and utilize a numerical scheme that is derived in parallel to the L1‐method for the time variable and a standard fourth‐order approximation in the spatial variable. The main method derived in this article has a rate of convergence of O ( k α + h 4 ) for u ( x , t ) ∈ C α ([0, T ]; C 6 (Ω)) , 0 < α < 1 , which improves previous regularity assumptions that require C 2 [0, T ] regularity in the time variable. We also present a novel alternative method for a first‐order approximation in time, under a regularity assumption of u ( x , t ) ∈ C 1 ([0, T ]; C 6 (Ω)) , while exhibiting order of convergence slightly more than O ( k ) in time. This allows for a much wider class of functions to be analyzed which was previously not possible under the L1‐method. We present numerical examples demonstrating these results and discuss future improvements and implications by using these techniques.