Fractional shifted legendre tau method to solve linear and nonlinear variable-order fractional partial differential equa
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ORIGINAL RESEARCH
Fractional shifted legendre tau method to solve linear and nonlinear variable‑order fractional partial differential equations Maliheh Shaban Tameh1 · Elyas Shivanian2 Received: 7 February 2020 / Accepted: 5 September 2020 © Islamic Azad University 2020
Abstract Here, we shed light on the fractional linear and nonlinear Klein–Gorden partial differential equations via Fractional Shifted Legendre Tau Method. With this objective, the operational matrices of fractional-order shifted Legendre functions (FSLFs) are derived and combined with the Tau method to convert the fractional-order differential equations to a system of solvable algebraic equations. The validity and the efficiency of the operational matrices are tested. Our findings yield an affirmative consequence, indicating applicability of the proposed method for nonlinear equations appearing in science and engineering. Keywords Klein–Gorden partial differential equations · Fractional-order Legendre functions · Operational matrices · Caputo fractional derivatives
Introduction The fractional differential equations and the fractional calculus have been widely used in the modeling and simulation of the problems in science and engineering such as electrochemistry, elastoplastic, thermoelastic and viscoelasticity. [1–4]. The fractional derivatives are of significant interest in the development of methods [5–11] to solve the fractional partial differential equations (FPDEs) such as Klein–Gordon equation [12] appearing in many scientific applications, particularly, solid-state physics and nonlinear optics [13]. Besides, shallow water wave equations [14] can be modeled via this equation. In the recent years, some analytical and numerical methods have been proposed to find the approximation of Klein–Gordon FPDEs [12, 15, 16]. In [1], a spectral collocation method is proposed that applies Legendre polynomials to approximate the solutions. Spectral methods are classified as Pseudospectral, Collocation, Galerkin and Tau methods. The Tau method [17–20] has been extensively used in a wide range of problems arising in the mathematical modelings to find the solution of the differential equations. * Elyas Shivanian [email protected] 1
Department of Chemistry, University of Minnesota, 55455, Minneapolis, MN, USA
Department of Applied Mathematics, Imam Khomeini International University, Qazvin 34149‑16818, Iran
2
In this method, appropriate basis functions that are typically the eigenfunctions of a singular Sturm-Liouville problem are used for the projection of the residual functions. The auxiliary conditions operate as constraints on the expansion coefficients. Legendre functions are categorized in the class of orthogonal functions. Various studies have proven the efficiency of these polynomials in fractional differential equations. Rida and Yousef, [21] proposed the fractional extension of the classical Legendre polynomials that replace the integer-order derivative in Rodrigues formula [22] by fractional-order derivatives. The known complexit
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