A high-order compact alternating direction implicit method for solving the 3D time-fractional diffusion equation with th
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ORIGINAL RESEARCH
A high‑order compact alternating direction implicit method for solving the 3D time‑fractional diffusion equation with the Caputo–Fabrizio operator Narjes Abdi1 · Hossein Aminikhah1,2 · Amir Hossein Refahi Sheikhani3 · Javad Alavi1 Received: 27 December 2019 / Accepted: 27 July 2020 © Islamic Azad University 2020
Abstract In this paper, a high-order compact finite difference method (CFDM) with an operator-splitting technique for solving the 3D time-fractional diffusion equation is considered. The Caputo–Fabrizio time operator is evaluated by the L1 approximation, and the second-order space derivatives are approximated by the compact CFDM to obtain a discrete scheme. Alternating direction implicit method (ADI) is used to split the problem into three separate one-dimensional problems. The local truncation error analysis is discussed. Moreover, the convergence and stability of the numerical method are investigated. Finally, some numerical examples are presented to demonstrate the accuracy of the compact ADI method. Keywords 3D time-fractional diffusion equation · Compact scheme · Finite difference method · Alternating direction implicit method · Caputo–Fabrizio operator
Introduction Fractional operators draw increasing attention due to its applications in many fields such as physics, biology, economics, signal and image processing, control, hydrology, and other studies [1–9]. This field of applied mathematics demonstrates the reality of nature better. Fractional differential equations are called differential equations, in which the order of derivative * Hossein Aminikhah [email protected] Narjes Abdi [email protected] Amir Hossein Refahi Sheikhani [email protected] Javad Alavi [email protected] 1
Department of Applied Mathematics and Computer Science, Faculty of Mathematical Sciences, University of Guilan, Rasht, Iran
2
Center of Excellence for Mathematical Modelling, Optimization and Combinational Computing (MMOCC), University of Guilan, Rasht, Iran
3
Department of Applied Mathematics, Faculty of Mathematical Sciences, Lahijan Branch, Islamic Azad University, Lahijan, Iran
operator is non-integer and they are a generalization of differential equations. The most important advantage of using fractional differential equations is their non-local property. It is well known that the integer-order differential operator is a local operator. It means that by local operator one can describe changes in a neighborhood of a point. The fractional-order differential operator is non-local. So the next status of a system depends not only on its current position but also depends on all of its past positions. Such type systems are called memory systems, and this is one reason why fractional calculus more popular [10]. Most fractional differential equations are very complicated, and it is difficult to obtain the exact solution; hence, ones obtain a numerical solution by a numerical method such as meshless local collocation method [11], implicit meshless method [12], matrix approach
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