Abstract
This paper deals with the initial value problem for linear systems of fractional differential equations (FDEs) with variable coefficients involving Riemann–Liouville and Caputo derivatives. Some basic properties of fractional derivatives and antiderivatives, including their non-symmetry w.r.t. each other, are discussed. The technique of the generalized Peano–Baker series is used to obtain the state-transition matrix. Explicit solutions are derived both in the homogeneous and inhomogeneous case. The theoretical results are supported by examples.
Highlights
Fractional differential equations (FDEs) provide a powerful tool to describe memory effect and hereditary properties of various materials and processes [1,2,3,4,5,6,7]
This paper deals with the initial value problem for linear systems of FDEs with variable coefficients involving Riemann–Liouville and Caputo derivatives
We examine homogeneous linear FDEs with variable coefficients involving Caputo derivatives
Summary
Fractional differential equations (FDEs) provide a powerful tool to describe memory effect and hereditary properties of various materials and processes [1,2,3,4,5,6,7]. Explicit solutions to linear systems of differential equations provide a basis to solve control problems. Analytical solutions of the linear systems of fractional differential equations with constant coefficients were derived in the papers [8,9] and applied to solving control problems and differential games in [10,11,12,13]. Only a few papers are devoted to solutions of the systems of FDEs with variable coefficients and their control. In [14], explicit solutions for the linear systems of initialized [15] FDEs are obtained in terms of generalized Peano–Baker series [16]. This paper deals with the initial value problem for linear systems of FDEs with variable coefficients involving Riemann–Liouville and Caputo derivatives.
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