An Efficient Spline Collocation Method for a Nonlinear Fourth-Order Reaction Subdiffusion Equation
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An Efficient Spline Collocation Method for a Nonlinear Fourth-Order Reaction Subdiffusion Equation Haixiang Zhang1 · Xuehua Yang1 · Da Xu2 Received: 27 April 2020 / Revised: 21 August 2020 / Accepted: 4 September 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract The nonlinear fourth-order reaction–subdiffusion equation whose solutions display a typical initial weak singularity is considered. A new analytical technique is introduced to analyze orthogonal spline collocation (OSC) method based on L1 scheme on graded mesh. By introducing a discrete convolution kernel and discrete fractional Grönwall inequality, convergence of the scheme is proved rigorously. This novel analytical technique can provide new insights in analyzing other time fractional fourth-order differential equations with weakly singular solutions. Keywords Fourth-order time fractional equation · Finite difference method · Collocation scheme · Convergence Mathematics Subject Classification 65N12 · 65N30 · 35K61
1 Introduction In the paper, we consider the following nonlinear fourth-order reaction–subdiffusion equation with initial singularity ⎧ α ⎨ ∂t u + Δ2 u = Δu + f (u) + g(x, t), x ∈ Ω, t ∈ (0, T ]; (1.1) u = u 0 (x), x ∈ Ω, t = 0; ⎩ u = Δu = 0, x ∈ ∂Ω, t ∈ (0, T ]. ¯ We assume that Ω has smooth Here, Ω ⊂ Rd (d = 1, 2). Its closure is denoted by Ω. ¯ boundary ∂Ω or is convex. u 0 ∈ C(Ω), g is the given function, the nonlinear function f (u)
The work was supported by National Natural Science Foundation of China (11701168, 11601144), Hunan Provincial Natural Science Foundation of China (2018JJ3108, 2018JJ3109, 2018JJ4062), Scientific Research Fund of Hunan Provincial Education Department (18B304, YB2016B033), and China Postdoctoral Science Foundation (2018M631403).
B
Xuehua Yang [email protected]
1
School of Science, Hunan University of Technology, Zhuzhou 412007, China
2
Department of Mathematics, Hunan Normal University, Changsha 410081, China 0123456789().: V,-vol
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Journal of Scientific Computing
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is smooth, and ∂tα u denotes the Caputo fractional derivative
∂tα u(x, t) =
0
t
ω1−α (t − s)
∂u(x, s) ds, ∂s
(1.2)
−α
where ω1−α (t − s) = (t−s) Γ (1−α) , 0 < α < 1, t > 0. The nonlinear equation (1.1) possesses the fractional sub-diffusion and fourth-order derivative terms simultaneously, which makes it distinctive compared to general timefractional sub-diffusion equations. For sub-diffusion equations with weakly singular solutions, their accurate numerical simulations have been the topic of much recent research, see references [1–5]. Yan et al. [6] established an improved L1 method for time fractional PDEs with nonsmooth data, then Xing and Yan modified this method to get a more higher order scheme in [7]. More recently, there are certain papers concerned with fourth-order fractional differential equations [8–13] and nonlinear sub-diffusions [14–19]. Ji et al. [20] proposed a high order FDM for fourth-order fractional sub-diffusion equations with the Dirichlet
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