A numerical method for solution of ordinary differential equations of fractional order
Leszczynski Jacek, Ciesielski Mariusz
TL;DR
This work tackles the challenge of solving fractional-order differential equations by introducing a decomposition approach that rewrites the problem as a system consisting of a standard integer-order ODE coupled with left inverse Abel-integral (inverse Abel) forms. Central to the method is the introduction of temporary variables such as $z_1(t)=(_{\tau}I^{m_1-\alpha_1}_t y)(t)$, which links the fractional derivatives $(_{\tau}D^{\alpha}_t y)(t)$ to integer-order dynamics via $(_{\tau}D^{\alpha}_t y)(t)=D_t^{m} z(t)$. The authors develop both one-term and multi-term classifications (including independent and dependent subclasses) and provide a concrete explicit numerical scheme that combines an integer-order stepping method with discretizations of the left-inverse Abel operators. They illustrate the approach with linear and nonlinear fractional models, notably the Bagley–Torvik equation, comparing results to analytical solutions where available and examining the influence of the time-step. The work offers a practical, general framework for numerically solving fractional ODEs by leveraging a structured decomposition that accommodates memory effects inherent in fractional dynamics.
Abstract
In this paper we propose an algorithm for the numerical solution of arbitrary differential equations of fractional order. The algorithm is obtained by using the following decomposition of the differential equation into a system of differential equation of integer order connected with inverse forms of Abel-integral equations. The algorithm is used for solution of the linear and non-linear equations.