Efficient methods for nonlinear time fractional diffusion-wave equations and their fast implementations
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Efficient methods for nonlinear time fractional diffusion-wave equations and their fast implementations Jianfei Huang1 · Dandan Yang2
· Laurent O. Jay3
Received: 9 October 2018 / Accepted: 23 September 2019 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Recently, numerous numerical schemes for solving linear time fractional diffusionwave equations have been developed. However, most of these methods require relatively high smoothness in time and need extensive computational work and large storage due to the nonlocal property of fractional derivatives. In this paper, an efficient scheme and an alternating direction implicit (ADI) scheme are constructed for one-dimensional and two-dimensional nonlinear time fractional diffusion-wave equations based on their equivalent partial integro-differential equations. The proposed methods require weaker smoothness in time compared to the methods based on discretizing fractional derivative directly. They are proved to be unconditionally stable and convergent with first-order of accuracy in time and second order of accuracy in space. Fast implementations of the proposed methods are presented by the sum-ofexponentials (SOE) approximation for the kernel t −2+α on the interval [τ, T ], where 1 < α < 2. Finally, numerical experiments are carried out to illustrate the theoretical results of our direct schemes and demonstrate their powerful computational performances. Keywords Time fractional diffusion-wave equations · Nonlinear system · Finite difference schemes · Stability · Convergence · Fast implementations Mathematics Subject Classification (2010) 65M06 · 65M12 · 35R11
Dandan Yang
[email protected] 1
College of Mathematical Sciences, Yangzhou University, Yangzhou 225002, China
2
School of Mathematical Science, Huaiyin Normal University, Huaian 223300, China
3
Department of Mathematics, 14 MacLean Hall, The University of Iowa, Iowa City, IA, 52242, USA
Numerical Algorithms
1 Introduction In this paper, we consider numerical methods for nonlinear time fractional diffusionwave problems of the following form = u(X, t) + f (X, t, u(X, t)), X ∈ , 0 < t ≤ T , 1 < α < 2, (1.1) with initial conditions C α 0 Dt u(X, t)
u(X, 0) = φ(X), ut (X, 0) = ϕ(X), X ∈ ,
(1.2)
and boundary condition u(X, t) = ψ(X, t), X ∈ ∂, 0 < t ≤ T ,
(1.3)
where the spatial variable X can be seen as the one-dimensional X = x or the twodimensional X = (x, y), is the Laplacian, is the domain of X, and ∂ and are the boundary and the closure of , respectively. f (X, t, u) is a nonlinear function of unknown u ∈ R and fulfills a Lipschitz condition with respect to u. φ(X), ϕ(X), α and ψ(X, t) are assumed to be sufficiently smooth functions. C 0 Dt u is the temporal Caputo derivative of order α defined as t ∂ 2 u(X, s) 1 C α D u(X, t) = (t − s)1−α ds. (1.4) 0 t
(2 − α) 0 ∂s 2 The time fractional diffusion-wave (1.1) possesses the remarkable feature that it can be considered as intermediate between parabolic diffusion equations and hyperbolic wave equations. It has
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