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ORIGINAL PAPER

Chebyshev spectral methods for multi-order fractional neutral pantograph equations S. S. Ezz-Eldien . Y. Wang . M. A. Abdelkawy . M. A. Zaky . A. A. Aldraiweesh . J. Tenreiro Machado

Received: 12 March 2019 / Accepted: 27 May 2020 Ó Springer Nature B.V. 2020

Abstract This paper is concerned with the application of the spectral tau and collocation methods to delay multi-order fractional differential equations with vanishing delay rx ð0\r\1Þ. The fractional derivatives are described in the Caputo sense. The model solution is expanded in terms of Chebyshev polynomials. The convergence of the proposed approaches is investigated in the weighted L2 -norm. Numerical examples are provided to highlight the convergence rate and the flexibility of this approach. Our results confirm that nonlocal numerical methods S. S. Ezz-Eldien (&)  Y. Wang Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China e-mail: [email protected] Y. Wang e-mail: [email protected] S. S. Ezz-Eldien  A. A. Aldraiweesh College of Education, King Saud University, Riyadh, Saudi Arabia e-mail: [email protected] S. S. Ezz-Eldien Department of Mathematics, Faculty of Science, New Valley University, El-Kharga 72511, Egypt

are best suited to discretize fractional differential equations as they naturally take the global behavior of the solution into account. Keywords Fractional differential equations  Pantograph equations  Spectral methods

M. A. Abdelkawy Department of Mathematics, Faculty of Science, BeniSuef University, Beni Suef, Egypt M. A. Zaky Department of Applied Mathematics, National Research Centre, Dokki, Giza 12622, Egypt e-mail: [email protected] J. T. Machado Department of Electrical Engineering, Institute of Engineering, Polytechnic of Porto, Rua Dr. Anto´nio Bernardino de Almeida, 431, 4249-015 Porto, Portugal e-mail: [email protected]

M. A. Abdelkawy Department of Mathematics and Statistics, College of Science, Al-Imam Mohammad Ibn Saud Islamic University, Riyadh, Saudi Arabia e-mail: [email protected]

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S. S. Ezz-Eldien et al.

1 Introduction

C a 0 Dx u

Fractional differential equations arise naturally in multifarious areas of science, engineering, and mathematics, see [10, 14, 21, 24]. There are various definitions of fractional derivatives. The two most important types are the Caputo and Riemann–Liouville definitions. A common feature of these derivatives, unlike the integer derivatives which are locally defined on the epsilon neighborhood of a chosen point, is that they are globally defined by a definite integral over the whole domain. Analytical methods have been used to obtain closed-form solutions to various kinds of fractional differential equations. Fractional variational iteration method [26] has been applied for timefractional Newell–Whitehead–Segel equation. Integral projected differential transform method [30] has been used for solving fractional wave equation. Fourier transform and