Atomic solution for both ordinary and fractional abstract Cauchy problem in tensor product form

In this paper, we discuss the atomic solutions of the abstract Cauchy problem in two variables. The proposed abstract Cauchy problem is assumed to be non-homogeneous and in tensor product form. We utilize some results from the theory of tensor product of Banach spaces to obtain solutions in the atom operator form. The approach is applied for both ordinary and α-conformable fractional derivatives.