The construction of a new operational matrix of the distributed-order fractional derivative using Chebyshev polynomials and its applications

Abstract

In this paper, the properties of Chebyshev polynomials and the Gauss–Legendre quadrature rule are employed to construct a new operational matrix of distributed-order fractional derivative. This operational matrix is applied for solving some problems such as distributed-order fractional differential equations, distributed-order time-fractional diffusion equations and distributed-order time-fractional wave equations. Our approach easily reduces the solution of all these problems to the solution of some set of algebraic equations. We also discuss the error analysis of approximation distributed-order fractional derivative by using this operational matrix. Finally, to illustrate the efficiency and validity of the presented technique five examples are given.

Abbreviations: DFDEs: distributed-order fractional differential equations; DTFDEs: distributed-order time-fractional diffusion equations; DTFWEs: distributed-order time-fractional wave equations; OMFD: operational matrix of fractional derivative; SCP: shifted Chebyshev polynomial

Keywords:

• Chebyshev polynomials
• distributed-order
• Caputo derivative
• fractional differential equations
• operational matrix

2010 AMS Subject Classifications:

• 26A33
• 65M70

Acknowledgments

The authors are very grateful to reviewers for their comments and suggestions which have led to improvement of the paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).