A Novel Operational Matrix of Caputo Fractional Derivatives of Fibonacci Polynomials: Spectral Solutions of Fractional Differential Equations

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.

A Discretization Approach for the Nonlinear Fractional Logistic Equation

“…Many other numerical and approximation methods as well as computational approaches have been developed and applied for the FDEs which are based upon various closely-related models of real-world problems. For example, Baleanu et al [ 23 ] made use of a Chebyshev spectral method based on operational matrices, a remarkable survey of numerical methods can be found in [ 24 ], a study of the fractional-order Bessel, Chelyshkov, and Legendre collocation schemes for the fractional Riccati equation was presented in [ 25 ], an operational matrix of fractional-order derivatives of Fibonacci polynomials was developed in [ 26 ], an introductory overview and recent developments involving FDEs was presented in [ 27 ], efficiency of the spectral collocation method in the dynamic simulation of the fractional-order epidemiological model of the Ebola virus was investigated in [ 28 ], the Jacobi collocation method and a spectral tau method based on shifted second-kind Chebyshev polynomilas for the approximate solution of some families of the fractional-order Riccati differential equations were discussed in [ 29 , 30 ], computational approaches to FDEs for the biological population model were discussed in [ 31 ], the generalized Chebyshev and Bessel colllocation approaches for fractional BVPs and multi-order FDEs were considered in [ 32 , 33 ], and a general wavelet quasi-linearization method for solving fractional-order population growth model was developed and applied in [ 34 ].…”

Fractional Time Derivative Seismic Wave Equation Modeling for Natural Gas Hydrate

“…In this paper we use the fractional time derivative wave equation to simulate the wave field. In recent years, many scholars have proposed different methods for solving fractional time derivatives [25,26], which have promoted its development.…”

Hermite collocation method for fractional order differential equations

“…Nowadays, many researchers are studying on numerical or approximate solutions of the fractional order equations and some new techniques have been introduced or adopted with the existent ones for ordinary case. For instance, finite difference [6][7][8][9][10], fractional linear multistep methods [11][12][13], Adomian decomposition [14][15][16], variational iteration method [16][17][18], differential transform or Taylor collocation method [19,20] and spectral method [21][22][23][24] can be cited here. For some classes of fractional differential equations, Kumar and Agarwal mentioned about polynomial approximation methods and detailed information can be found in [25][26][27].…”