Solving nonlinear systems of fractional-order partial differential equations using an optimization technique based on ge
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Solving nonlinear systems of fractional-order partial differential equations using an optimization technique based on generalized polynomials H. Hassani1 · J. A. Tenreiro Machado2 · E. Naraghirad3 · B. Sadeghi4 Received: 18 April 2020 / Revised: 27 August 2020 / Accepted: 13 October 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract This paper addresses the application of generalized polynomials for solving nonlinear systems of fractional-order partial differential equations with initial conditions. First, the solutions are expanded by means of generalized polynomials through an operational matrix. The unknown free coefficients and control parameters of the expansion with generalized polynomials are evaluated by means of an optimization process relating the nonlinear systems of fractionalorder partial differential equations with the initial conditions. Then, the Lagrange multipliers are adopted for converting the problem into a system of algebraic equations. The convergence of the proposed method is analyzed. Several prototype problems show the applicability of the algorithm. The approximations obtained by other techniques are also tested confirming the high accuracy and computational efficiency of the proposed approach. Keywords Nonlinear system of fractional-order partial differential equations · Generalized polynomials · Operational matrix · Optimization problem · Control parameters Mathematics Subject Classification 35R11 · 35G50 · 35C10
Communicated by Agnieszka Malinowska.
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H. Hassani [email protected] J. A. Tenreiro Machado [email protected] E. Naraghirad [email protected] B. Sadeghi [email protected]
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Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam
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Department of Electrical Engineering, Institute of Engineering, Polytechnic of Porto, R. Dr. António Bernardino de Almeida 413, 4249-015 Porto, Portugal
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Department of Mathematics, Yasouj University, Yasouj, Iran
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Department of Computer Engineering, Payame Noor University, Tehran, Iran 0123456789().: V,-vol
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1 Introduction Fractional calculus generalizes the differentiation and integration to non-integer orders (Podlubny 1999). In recent years, fractional differential equations attracted the attention of researchers in many branches of science such as mathematics, physics, engineering, and biosciences (Miller and Ross 1993; Hilfer 2000; Zhou et al. 2015; Hassani et al. 2019, 2020; Riewe 1997; Shekari et al. 2019; Rahimkhani and Ordokhani 2020; Zhao et al. 2019). The modeling of systems with differential equations has been an important research topic. Owolabi et al. (2018) studied the analytical and numerical solutions of a dynamical model comprising three species of systems using the fractional Fourier transform. Alliera and Amster (2018) used the topological degree theory and proved the existence of positive periodic solutions for a system of delay differential equations. Ablinger et al. (2019) develop
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