Abstract
A new direct operational inversion method is introduced for solving coupled linear systems of ordinary fractional differential equations. The solutions so‐obtained can be expressed explicitly in terms of multivariate Mittag‐Leffler functions. In the case where the multiorders are multiples of a common real positive number, the solutions can be reduced to linear combinations of Mittag‐Leffler functions of a single variable. The solutions can be shown to be asymptotically oscillatory under certain conditions. This technique is illustrated in detail by two concrete examples, namely, the coupled harmonic oscillator and the fractional Wien bridge circuit. Stability conditions and simulations of the corresponding solutions are given.
Highlights
Fractional differential equations are well suited to model physical systems with memory or fractal attributes
In this paper we introduce a direct operational method to solve a system of linear inhomogeneous CFDEs
Subsequent sections deal with the applications of the coupled fractional differential equations to two physical systems, namely, the coupled fractional oscillator and the fractional Wien bridge circuit, as examples to illustrate the proposed method
Summary
Fractional differential equations are well suited to model physical systems with memory or fractal attributes. In this paper we introduce a direct operational method to solve a system of linear inhomogeneous CFDEs. We will restrict our discussion to a system of linear nonhomogeneous ordinary differential equations of arbitrary fractional-orders. We will restrict our discussion to a system of linear nonhomogeneous ordinary differential equations of arbitrary fractional-orders These equations based on two types of fractional derivatives will be considered, namely, the Caputo and RiemannLiouville fractional derivatives. When each order of the CFDEs is an integer multiple of a certain common real positive number, it is possible to further reduce the solutions to the single-variate Mittag-Leffler functions. For such cases, we study the conditions for the existence of asymptotic periodic solutions. Subsequent sections deal with the applications of the coupled fractional differential equations to two physical systems, namely, the coupled fractional oscillator and the fractional Wien bridge circuit, as examples to illustrate the proposed method
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