Representation of solutions to tempered delayed ψ-fractional systems with noncommutative coefficients

This paper focuses on deriving explicit solutions for tempered delayed fractional differential systems that utilize Caputo fractional derivatives in relation to another function. To achieve this, we define tempered $\psi$-delayed perturbations of Mittag-Leffler type functions and explore their $\psi$-Laplace transforms. Additionally, we discuss theorems related to shifting and time-delay in the context of $\psi$-Laplace transforms. Utilizing the tempered $\psi$-delayed perturbational function, we establish a representation of explicit solutions for the system through the Laplace transform method. This representation is validated by demonstrating that it satisfies the system, alongside employing the method of variation of constants. Several novel special cases are introduced, and a numerical example is provided to demonstrate the practical application of the results obtained.