Two linearized schemes for time fractional nonlinear wave equations with fourth-order derivative
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Two linearized schemes for time fractional nonlinear wave equations with fourth-order derivative Jianfei Huang1 · Zhi Qiao1 · Jingna Zhang1 · Sadia Arshad2 · Yifa Tang3,4 Received: 2 May 2020 / Revised: 7 October 2020 / Accepted: 21 October 2020 © Korean Society for Informatics and Computational Applied Mathematics 2020
Abstract In this paper, two linearized schemes for time fractional nonlinear wave equations (TFNWEs) with the space fourth-order derivative are proposed and analyzed. To reduce the smoothness requirement in time, the considered TFNWEs are equivalently transformed into their partial integro-differential forms by the Riemann–Liouville integral. Then, the first scheme is constructed by using piecewise rectangular formulas in time and the fourth-order approximation in space. And, this scheme can be fast evaluated by the sum-of-exponentials technique. The second scheme is developed by using the Crank–Nicolson technique combined with the second-order convolution quadrature formula. By the energy method, the convergence and unconditional stability of the proposed schemes are proved rigorously. Finally, numerical experiments are given to support our theoretical results. Keywords Fractional nonlinear wave equations · Fourth-order derivative · Linearized schemes · Stability · Convergence Mathematics Subject Classification 65M06 · 65M12
1 Introduction Fractional partial differential equations (FPDEs) play an increasingly important role in modeling of anomalous phenomena and complex systems. FPDEs with fractional
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Jianfei Huang [email protected]
1
College of Mathematical Sciences, Yangzhou University, Yangzhou 225002, China
2
COMSATS University Islamabad, Lahore Campus, Islamabad, Pakistan
3
LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
4
School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
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J. Huang et al.
derivatives in space can represent large particle jumps, while long waiting time is associated with fractional derivatives in time, see [15,24,35,42] for examples. There is a large number of works devoting to obtain the analytic solutions for FPDEs by using some analytical mathematical tools, such as Fourier transform, Laplace transform and Mellin transform, see [36,44]. However, the closed-form solutions have been only found for a small class of FPDEs, it is still difficult or impossible to obtain the analytic solutions of FPDEs, especially the nonlinear ones. Thus, there has been a growing interest to develop and analyze numerical methods for solving FPDEs or related problems, see [2,4–8,13,19,26,27,30,33,37–40] and the references therein. In this paper, we will consider the following time fractional nonlinear wave equations with the fourth-order derivative in space [1], ∂ 4 u(x, t) = f (x, t) + g(u), 0 < t ≤ T , ∂x4 u(x, 0) = 0, u t (x, 0) = 0, 0 < x < L, u(0, t) = u(L, t) = u x (0, t) = u x (L, t) = 0, 0 < t ≤ T ,
C α 0 Dt u(x, t) +
(1)
where f (x, t) is a known function,
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