A numerical approach for solving a class of variable-order fractional functional integral equations

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A numerical approach for solving a class of variable-order fractional functional integral equations Farzad Khane Keshi1 · Behrouz Parsa Moghaddam1 Arman Aghili2

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Received: 29 October 2017 / Revised: 24 February 2018 / Accepted: 28 February 2018 / Published online: 7 March 2018 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2018

Abstract This paper proposes new discretization techniques to estimate variable-order fractional integral operators based on the piecewise integro quadratic spline interpolation. The proposed methods are modified to solve a class of variable-order fractional functional integral equations. Moreover, we investigate the performance of the proposed methods by solving the variable-order fractional pantograph and Emden–Fowler functional integral equations. Keywords Variable-order fractional calculus · Integro spline · Richardson extrapolation · Discretization error · Pantograph functional integral equation · Emden–Fowler functional integral equation Mathematics Subject Classification 46N20 · 65Q20 · 26A33 · 34K28 · 65L70

1 Introduction The research on fractional operators has attracted much scholarly attention due to their unique non-local properties (Heymans and Bauwens 1994; Hilfer 2000; Dabiri et al. 2016; Hajipour et al. 2017). The advantages of using fractional operators in describing processes with memory or hereditary property (Dabiri et al. 2016, 2017; Baleanu et al. 2017) and

Communicated by Vasily E. Tarasov.

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Behrouz Parsa Moghaddam [email protected] Farzad Khane Keshi [email protected] Arman Aghili [email protected]

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Department of Mathematics, Lahijan Branch, Islamic Azad University, Lahijan, Iran

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Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Guilan, Rasht, Iran

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the study of creep or relaxation in visco-elastoplastic materials (Bagley and Torvik 1986), diffusion process models (Samko et al. 1993), plasma physics (Samko et al. 1993), or control problems (Das 2011; Dabiri et al. 2016a, b). In the last decade, a category of fractional calculus called variable-order (VO) fractional calculus has been used to describe hereditary processes in a more efficient way in comparison to fixed order (FO) fractional calculus. Furthermore, the advantages of using VO fractional calculus compared to FO fractional calculus have been revealed in various disciplines such as modeling of many mechanical, electrical, physiological and hormonal control systems (Lorenzo and Hartley 2002; Coimbra 2003; Kobelev et al. 2003; Pedro et al. 2008; Diaz and Coimbra 2008; Sun et al. 2009). It is often impossible to derive an analytical solution for VO fractional problems owing to the definitions of VO fractional operators, and hence several computational techniques have been suggested to treat this problem. The most commonly proposed approximations are based on the finite difference methods (Sun et al. 2012; Shen et al. 2012, 2013; Chen et al. 2012; Zhang et al. 2013; Moghaddam and Machado 2017a), spec

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