International Journal of Neutrosophic Science IJNS 2690-6805 2692-6148 10.54216/IJNS https://www.americaspg.com/journals/show/4096 2020 2020 Department of Mathematics, Al Zaytoonah University of Jordan, Amman, Jordan Waseem Waseem In this paper, we utilize the concept of point-wise independent set of closed operators that enabled us to find atomic solutions of the non-homogeneous α−fractional abstract Cauchy problem of order n. The proposed fractional abstract Cauchy problem is Anu(nα)(t) + An−1u((n−1)α)(t) + · · · + A1u(α)(t) + A◦u(t) = f (t) where the involved operators An, An−1, · · · , A◦ are closed and linear on a given Banach space and the unknown function u(t) is assumed to be n-times α−differentiable. Beyond the deterministic setting, we indicate how the atomic-solution framework extends naturally when coefficients, data, or initial states are modeled as neutrosophic (single-valued) quantities, thereby accommodating uncertainty and indeterminacy at the operator or forcing level. 2026 2026 407 412 10.54216/IJNS.270233 https://www.americaspg.com/articleinfo/21/show/4096