Exact solutions and Hyers-Ulam stability of fractional equations with double delays

In this paper, we discuss the exact solutions of linear homogeneous and nonhomogeneous fractional differential equations with double delays. Firstly, a new concept of double-delayed Mittag-Leffler type matrix function is introduced, which is the promotion of the double-delayed matrix exponential. Secondly, we apply the double-delayed Mittag-Leffler type matrix function and Laplace transform approach to obtain the exact solutions of fractional differential equations with double delays. Furthermore, the solution is used to investigate the Hyers-Ulam stability of the system. Lastly, we illustrate our techniques by an example.

Automated summary

This paper discusses the exact solutions of linear homogeneous and nonhomogeneous fractional differential equations with double delays. A new concept of double-delayed Mittag-Leffler type matrix function is introduced, and it is applied along with Laplace transform approach to obtain the exact solutions. The solutions are also used to investigate the Hyers-Ulam stability of the system, and an example is provided to illustrate the techniques.