Finite rank solution for conformable higher-order abstract Cauchy problem in Hilbert space

We established existence and uniqueness of finite-rank solutions to the conformable fractional abstract Cauchy problem of arbitrary order $n\alpha$ ( $n\ge 2$, $0<\alpha \le 1$) in Hilbert space ${\ell }^2$. The problem has the form

$$E{u}^{(n\alpha )}(t)+A_{n-1}{u}^{((n-1)\alpha )}(t)+\cdots +A_{0}u(t)=f(t)z,$$

with closed linear operators $A_k$, $E$ on ${\ell }^2$. Using tensor product decomposition and a companion-type reduction to a first-order conformable system, we proved the result for both the nondegenerate case ( $E=I$) and the degenerate case ( $E\ne I$, orthogonally diagonalizable with $A_{n-1}{|}_{\ker (E)}$ invertible) via strong induction on $n$. The inverse problem, in which $u(t)$ and $f(t)$ are both unknown, was solved uniquely under an inner-product measurement condition. Applications to viscoelastic oscillator models were given, including a Kelvin–Voigt fractional oscillator, a multi-mode viscoelastic beam, and a partially constrained degenerate system, with explicit numerical solutions and graphical illustrations. The case $n=2$ recovered previously established results as a special case.