Abstract
An operator-based scheme for the numerical integration of fractional differential equations is presented in this paper. The generalized differential operator is used to construct the analytic solution to the corresponding characteristic ordinary differential equation in the form of an infinite power series. The approximate numerical solution is constructed by truncating the power series, and by changing the point of the expansion. The developed adaptive integration step selection strategy is based on the controlled error of approximation induced by the truncation. Computational experiments are used to demonstrate the efficacy of the proposed scheme.
Highlights
Fractional differential equations (FDEs) play an important role in many research fields
The main objective of this paper is to present a novel FDE integration scheme based on the generalized differential operator technique
A novel semi-analytical scheme for the numerical integration of fractional differential equations was presented in this paper
Summary
Fractional differential equations (FDEs) play an important role in many research fields. A new scheme for the construction of numerical solutions that can be applied to several types of fractional derivatives is discussed in [11]. An approach based on Chebyshev polynomials with time-dependent coefficients is employed to construct numerical solutions to Caputo-type time–space fractional partial differential equations with variable coefficients in [13]. The main objective of this paper is to present a novel FDE integration scheme based on the generalized differential operator technique. Riccati-type equations have recently been discussed in a number of publications concerned with presenting novel FDE integration schemes. A modification of the homotopy perturbation method is used in [22] to construct numerical solutions to the Riccati-type FDEs. As an example, the following form of the fractional Riccati equation is considered in developing the numerical FDE integration strategy:.
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