Properties of solutions for fractional-order linear system with differential equations

Abstract

In this paper, we study the analytical solutions of two-dimensional fractional-order linear system $ \mathcal{D}^{\alpha}_{t}X(t) = AX(t) $ described by fractional differential equations, where $ \mathcal{D} $ is the fractional derivative in the Caputo-Fabrizio sense and $ A = (a_{ij})_{2\times2} $ is nonsingular coefficient matrix with $ a_{ij}\in\mathbb{R} $. The analytical solutions of fractional-order linear system will be compared to the solution of classical linear system. Examples are provided to characterize the behavior of the solutions for fractional-order linear system.

Keywords

• linear system
• caputo-fabrizio derivative
• analytical solution
• fractional differential
• fractional integral