Analytic Solution of Higher Order Fractional Abstract Cauchy
Problem
Waseem Ghazi Alshanti1,∗
1Department of Mathematics, Al Zaytoonah University of Jordan, Amman, Jordan
Email: w.alshanti@zuj.edu.jo
Abstract
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 un-
known 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.
Keywords: α-fractional Abstract Cauchy problem of order n; Pointwise independent operators; Atomic solu-
tion; Conformable fractional derivative