Spectral collocation method for nonlinear Caputo fractional differential system
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Spectral collocation method for nonlinear Caputo fractional differential system Zhendong Gu1 Received: 18 February 2020 / Accepted: 9 July 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract A spectral collocation method is developed to solve a nonlinear Caputo fractional differential system. The main idea is to solve the corresponding system of weakly singular nonlinear Volterra integral equations (VIEs). The convergence analysis in matrix form shows that the presented method has spectral convergence. Numerical experiments are carried out to confirm theoretical results. Keywords Spectral collocation method · Fractional differential system · Caputo · Convergence analysis · Numerical experiments Mathematics Subject Classification (2010) 65M70 · 45D05
1 Introduction Over the last few decades, engineers and scientists have developed new models that involve fractional differential equations (FDEs). These models have been applied successfully, e.g., in mechanics, (bio-)chemistry, electrical engineering, and medicine [5, 12, 23] and references therein. Many numerical methods are proposed to solve FDEs, such as Adams-Bashforth-Moulton method [5], perturbation-iteration method [15], hybrid spectral element method [30], spectral method [10, 20], Tau collocation method [1], finite difference method [18, 31, 33], finite elements method [13, 14, 17, 24, 32, 36, 41], collocation method [16, 21, 22, 27], spline collocation method [26], Chebyshev operational matrix method [2], predictor corrector method [6], and decomposition method [25, 29].
Communicated by: Martin Stynes Zhendong Gu
[email protected]; [email protected]; [email protected] 1
School of Financial Mathematics and Statistic, Guangdong University of Finance, Guangzhou 510521, China
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Adv Comput Math
(2020) 46:66
Spectral methods are the high accuracy numerical methods. They were developed to solve FDEs. Li, Zeng, and Liu [19] investigated the spectral approximations to the fractional integral and derivative. Zayernouri and Karniadakis [40] developed a fractional spectral collocation method for solving steady-state and time-dependent fractional PDEs. Chen, Shen, and Wang [4] proposed a spectral approximation of fractional differential equations by defining a new class of generalized Jacobi functions. In [34], a spectral collocation method was developed for nonlinear fractional boundary value problems with a Caputo derivative of order α ∈ (1, 2). In [37], spectral collocation methods were proposed to solve nonlinear multi-order FDEs by a suitable smoothing transformation. In [38], a spectral collocation method was used to approximate solutions of nonlinear multi-order fractional initial value problems with smooth solutions. In [39], a Jacobi spectral method was developed to solve nonlinear systems of single-term fractional boundary value problems and related VolterraFredholm integral equations with smooth solutions. The Jacobi spectral collocation method [9] was developed to solve linear multi-order FDEs by using a vari
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