Abstract
We extend several fractional Laplace transform results to matrix-valued functions in this paper, which we will utilize to obtain some useful and valuable theorems, like results on the set of the piecewise continuous functions with conformable exponential order and the low and conditions to obtain the fractional Laplace transform of matrix-valued functions. As an application, we apply the obtained theorems to solve certain fractional initial value problems for vector-valued and matrix-valued functions and the solution will be victor or matrix and it will exact. Also, we establish a fractional system transfer matrix and the conformable fractional Laplace transform of the conformable exponential matrix function, which will give us directly the exact solution for a precisely type of fractional initial value problems.
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