Fractional Linear Systems With Nonsingular Kernel: Explicit Solutions and Ulam–Hyers Stability

This paper is devoted to the explicit representation of solutions for a class of differential equations involving the Caputo–Fabrizio fractional derivative with a nonsingular kernel. Motivated by the lack of explicit representation formulas for Caputo–Fabrizio fractional systems in the existing literature, two distinct approaches based on Laplace transform techniques and variation of constants are developed to derive closed-form solution formulas. Furthermore, composition properties between the Caputo–Fabrizio fractional integral and derivative are established without imposing exponential boundedness assumptions. By employing the classical Gronwall inequality, the exponential boundedness of solutions is investigated, and the behavior of the corresponding Caputo–Fabrizio fractional derivative is consequently discussed. In addition, the Ulam–Hyers stability of the considered system is analyzed. Finally, an application to LR electrical circuits is presented to illustrate the applicability of the obtained theoretical results and the role of nonsingular memory effects in fractional electrical models.