The novel operational matrices based on 2D-Genocchi polynomials: solving a general class of variable-order fractional pa
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The novel operational matrices based on 2D-Genocchi polynomials: solving a general class of variable-order fractional partial integro-differential equations Haniye Dehestani1
· Yadollah Ordokhani1
· Mohsen Razzaghi2
Received: 5 April 2020 / Revised: 19 August 2020 / Accepted: 21 August 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract The main purpose of this study was to introduce an efficient approximate approach for solving a general class of variable-order fractional partial integro-differential equations (VOFPIDEs). First, the Genocchi polynomials properties and the pseudo-operational matrix of the VO-fractional derivative and fractional integration are presented. Then, to approximate an integral part of the problems, we obtain the dual pseudo-operational matrix of fractional order with a new technique. The pseudo-operational matrices of fractional order and Genocchi collocation method are applied to reduce the VO-FPIDEs to a system of algebraic equations. The error estimation indicates that the approximate solution converges to the exact solution. Also, we discuss in detail on the upper bound of error for the pseudo-operational matrix of fractional integration. In addition, to illustrate the efficiency and applicability of the approach, we present several numerical examples. Keywords Genocchi polynomials · Fractional pseudo-operational matrix · Variable-order fractional partial integro-differential equations · Mixed Riemann–Liouville integral · Variable-order fractional derivative Mathematics Subject Classification 26A33 · 33F05 · 35R09
Communicated by Agnieszka Malinowska.
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Yadollah Ordokhani [email protected] Haniye Dehestani [email protected] Mohsen Razzaghi [email protected]
1
Department of Mathematics, Faculty of Mathematical Sciences, Alzahra University, Tehran, Iran
2
Department of Mathematics and Statistics, Mississippi State University, Starkville, MS 39762, USA 0123456789().: V,-vol
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1 Introduction Partial integro-differential equations play an important role in various fields of engineering and science such as plasma physics (Meleshko et al. 2010), electromagnetic theory (Bloom 1981), nuclear reactor dynamics (Pachpatte 1983; Pao et al. 1979; Pao 1974), geophysics (Grasselli et al. 1990), reaction diffusion problems (Engler 1983), poro-viscoelastic media (Habetler and Schiffman 1970). Many researchers are investigating various forms of partial integro-differential equations and constructed numerous analytical and numerical methods. Express some of these methods: Spline collocation methods (Greenwell-Yanik and Fairweather 1986; Yan and Fairweather 1992), discontinuous Galerkin method (Larsson et al. 1998), finite element methods (Yanik and Fairweather 1988; Ma 2007), radial basis function-based differential quadrature method (Shu et al. 2004), Homotopy perturbation method (Babolian and Dastani 2011), differential transform method (Jang 2009), and there are other methods that readers
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