Inverse initial problem for fractional wave equation with the Hadamard fractional derivative

This paper investigates an inverse initial problem for a time-fractional wave equation involving the Hadamard fractional derivative. Unlike the more widely studied Caputo and Riemann–Liouville derivatives, the Hadamard derivative is defined via a logarithmic kernel and exhibits distinct analytical features, making it suitable for modeling processes with slow memory decay and multiplicative structures. The study is motivated by recent advances in fractional calculus and the growing interest in fractional models across applied sciences. Building on prior work concerning the extremum principle and solvability of boundary value problems with Hadamard-type operators, we establish sufficient conditions for the unique solvability of the inverse problem. The analysis is carried out in terms of eigenfunction expansions and leverages properties of the two-parameter Mittag–Leffler function. The findings contribute to the theory of inverse problems for fractional wave equations and highlight the role of Hadamard derivatives in capturing complex temporal dynamics in mathematical models.