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A new operational matrix of fractional order integration for the Chebyshev wavelets and its application for nonlinear fractional Van der Pol oscillator equation

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Abstract

In this paper, an efficient and accurate computational method based on the Chebyshev wavelets (CWs) together with spectral Galerkin method is proposed for solving a class of nonlinear multi-order fractional differential equations (NMFDEs). To do this, a new operational matrix of fractional order integration in the Riemann–Liouville sense for the CWs is derived. Hat functions (HFs) and the collocation method are employed to derive a general procedure for forming this matrix. By using the CWs and their operational matrix of fractional order integration and Galerkin method, the problems under consideration are transformed into corresponding nonlinear systems of algebraic equations, which can be simply solved. Moreover, a new technique for computing nonlinear terms in such problems is presented. Convergence of the CWs expansion in one dimension is investigated. Furthermore, the efficiency and accuracy of the proposed method are shown on some concrete examples. The obtained results reveal that the proposed method is very accurate and efficient. As a useful application, the proposed method is applied to obtain an approximate solution for the fractional order Van der Pol oscillator (VPO) equation.

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Authors and Affiliations

  1. Department of Mathematics, Shiraz University of Technology, Shiraz, Iran

    M H Heydari

  2. Center of Excellence for Robust Intelligent Systems, Yazd University, Yazd, Iran

    M H Heydari & M R Hooshmandasl

  3. The Laboratory of Quantum Information Processing, Yazd University, Yazd, Iran

    M H Heydari & M R Hooshmandasl

  4. Department of Computer Science, Yazd University, Yazd, Iran

    M R Hooshmandasl

  5. Engineering School (DEIM), University of Tuscia, Viterbo, Italy

    C Cattani

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  1. M H Heydari
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Correspondence to M R Hooshmandasl.

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Communicating Editor: Neela Nataraj

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Heydari, M.H., Hooshmandasl, M.R. & Cattani, C. A new operational matrix of fractional order integration for the Chebyshev wavelets and its application for nonlinear fractional Van der Pol oscillator equation. Proc Math Sci 128, 26 (2018). https://doi.org/10.1007/s12044-018-0393-4

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  • DOI: https://doi.org/10.1007/s12044-018-0393-4

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