Numerical solution of space fractional diffusion equation by spline method combined with Richardson extrapolation
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Numerical solution of space fractional diffusion equation by spline method combined with Richardson extrapolation Z. Soori1 · A. Aminataei2 Received: 27 December 2019 / Revised: 14 March 2020 / Accepted: 8 April 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract In this paper, we propose a high-order method for numerical solution of space fractional diffusion equation (SFDE) in one and two dimensions. The space fractional derivative of order 1 < α < 2 is described in Caputo’s sense. The spline approximation of Caputo is considered which has second-order accuracy in space. To improve the spatial accuracy, Richardson extrapolation method is presented. As a result, the high-order method can be viewed as the modification of the existing jobs (Sousa, Comput Math Appl 62:938–944, 2011; Salehi et al., Appl Math Comput 336:465–480, 2018). For the two-dimensional case, an alternating direction implicit (ADI) scheme is considered to split the equation into two separate one-dimensional equations. Moreover, the proposed scheme is extended to twosided SFDE case. Numerical results confirm the theoretical supports and the effectiveness of the proposed scheme. Keywords Caputo fractional derivative · High-order numerical scheme · Space fractional diffusion equation · Spline approximation · Richardson extrapolation · Two-sided space fractional diffusion equation Mathematics Subject Classification 26A33 · 41A15 · 65L06
1 Introduction In recent years, fractional calculus has played an important role in many fields of physics, chemistry, mechanics, electricity, signal and processing, etc. (Guo et al. 2011; Herrmann 2014; Hilfer 2000; Podlubny 1999; Povstenko 2015; Uchaikin 2013). In most existing literature, much attention is paid to the high-order methods for the SFDE as follows:
Communicated by José Tenreiro Machado.
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Z. Soori [email protected] A. Aminataei [email protected]
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Faculty of Mathematics, K. N. Toosi University of Technology, Tehran, Iran
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P.O. Box: 1676-53381, Tehran, Iran 0123456789().: V,-vol
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⎧ ∂u(x,t) ⎨ ∂t = d(x)Ca Dxα u(x, t) + p(x, t), u(x, 0) = φ(x), ⎩ u(a, t) = ϕ1 (t), u(b, t) = ϕ2 (t),
Z. Soori and A. Aminataei
x ∈ , 0 ≤ t ≤ T , x ∈ = ∪ ∂, 0 < t ≤ T,
(1)
where = (a, b), ∂ is the boundary of , p(x, t) is a known function, φ(x), ϕ1 (t) and ϕ2 (t) are given continuous functions and C0 Dxα is the α-th Caputo fractional derivative defined as x 2 ∂ u(ξ, t) 1 C α (x − ξ )1−α dξ, 1 < α < 2. (2) a Dx u(x, t) = (2 − α) a ∂ξ 2 Up to now, there have been existed several numerical methods to the SFDE. Author (Khader 2011) considered the Chebyshev approximations for reducing SFDE to a system of ordinary differential equations. They used the finite difference method to solve the resulting system. Wang and Basu (2012) developed a fast algorithm for two-dimensional SFDE based on the coefficient matrix of the traditional finite difference method which requires work count of N log N per iteration. A class of fourth-order approximation ba
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