@article{Odetallah2026, 
author = {Huda Odetallah and Mayada Abualhomos and Husam Miqdad and Tala Sasa and Lubaba Shaikh and Raja'a Al-Naimi},
title = {Finite rank solution for conformable higher-order abstract Cauchy problem in Hilbert space},
year = {2026},
journal = {AIMS Mathematics},
volume = {11},
number = {7},
pages = {21153-21166},
keywords = {abstract Cauchy problem, conformable fractional derivative, Higher-order, tensor product of Banach spaces, finite-rank function, viscoelastic oscillator},
url = {https://www.sciopen.com/article/10.3934/math.2026859},
doi = {10.3934/math.2026859},
abstract = {We established existence and uniqueness of finite-rank solutions to the conformable fractional abstract Cauchy problem of arbitrary order    n  α (   n  ≥  2,    0  <  α  ≤  1) in Hilbert space        ℓ          2      . The problem has the form     E        u          (      n      α      )        (  t  )  +      A          n      −      1              u          (      (      n      −      1      )      α      )        (  t  )  +  ⋯  +      A    0      u  (  t  )  =  f  (  t  )  z  ,with closed linear operators        A    k  ,    E on        ℓ          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  ≠  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.}
}