Abstract
Herein, two numerical algorithms for solving some linear and nonlinear fractional-order differential equations are presented and analyzed. For this purpose, a novel operational matrix of fractional-order derivatives of Fibonacci polynomials was constructed and employed along with the application of the tau and collocation spectral methods. The convergence and error analysis of the suggested Fibonacci expansion were carefully investigated. Some numerical examples with comparisons are presented to ensure the efficiency, applicability and high accuracy of the proposed algorithms. Two accurate semi-analytic polynomial solutions for linear and nonlinear fractional differential equations are the result.
Highlights
Fractional calculus is a vital branch of mathematical analysis which deals with derivatives and integrals to an arbitrary order
Many physical and engineering models are described by fractional differential equations
Chebyshev polynomials of the second kind were employed by Sweilam et al for solving space fractional-order diffusion equations [10]
Summary
Fractional calculus is a vital branch of mathematical analysis which deals with derivatives and integrals to an arbitrary order (real or complex). A number of articles were devoted to introducing numerical solutions for these equations In this respect, a Legendre operational matrix of fractional derivatives were constructed and employed for solving some types of FDEs in [19]. The principal objective of the present article is to present and implement new numerical spectral solutions of some types of FDEs. The principal objective of the present article is to present and implement new numerical spectral solutions of some types of FDEs For this purpose, the operational matrix of fractional derivatives of Fibonacci polynomials are constructed, and employed along with the tau and collocation spectral methods for obtaining numerical solutions of FDEs. The paper is organized as follows: first, in Section 2, some preliminaries including some fundamental definitions of the fractional calculus theory are presented.
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