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.

Keywords

• Fibonacci polynomials
• operational matrix
• spectral methods
• modified Bessel functions
• fractional-order differential equations
• Van der Pol oscillator
• Rayleigh equation