Abstract
In this article, we develop a numerical method based on the operational matrices of shifted Vieta–Lucas polynomials (VLPs) for solving Caputo fractional-order differential equations (FDEs). We derive a new operational matrix of the fractional-order derivatives in the Caputo sense, which is then used with spectral tau and spectral collocation methods to reduce the FDEs to a system of algebraic equations. Several numerical examples are given to show the accuracy of this method. These examples show that the obtained results have good agreement with the analytical solutions in both linear and non-linear FDEs. In addition to this, the numerical results obtained by using our method are compared with the numerical results obtained otherwise in the literature.
Highlights
Fractional calculus has been playing a very important role in scientific computations.Scientists are able to describe and model many physical phenomena with fractional-order differential equations
Motivated by the aforementioned works, we extend the study of the spectral methods by constructing a numerical algorithm that is based on the fractional-order derivative operational matrix of Vieta–Lucas polynomials (VLPs) in Caputo sense, together with the spectral tau method and spectral collocation method
It is important to mention that the proposed algorithm is computer-oriented and is capable of reducing the fractional-order differential equations (FDEs) to a system of algebraic equations, which greatly simplifies the problems
Summary
Fractional calculus has been playing a very important role in scientific computations. Dehestani et al [31] presented a novel collocation method based on the Genocchi wavelet for the numerical solution of FDEs and time-fractional partial differential equations with delay. Motivated by the aforementioned works, we extend the study of the spectral methods by constructing a numerical algorithm that is based on the fractional-order derivative operational matrix of VLPs in Caputo sense, together with the spectral tau method and spectral collocation method. The derivative terms are approximated by using the fractional-order derivative operational matrix of VLPs. It is important to mention that the proposed algorithm is computer-oriented and is capable of reducing the FDEs to a system of algebraic equations, which greatly simplifies the problems.
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