Volume 27 • Issue 2 • PP: 407-412 • 2026
Analytic Solution of Higher Order Fractional Abstract Cauchy Problem
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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.
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