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

Huda Odetallah; Mayada Abualhomos; Husam Miqdad; Tala Sasa; Lubaba Shaikh; Raja'a Al-Naimi; Huda Odetallah; Mayada Abualhomos; Husam Miqdad; Tala Sasa; Lubaba Shaikh; Raja'a Al-Naimi

doi:10.3934/math.2026859

AIMS Mathematics

2026, Volume 11, Issue 7: 21153-21166. doi: 10.3934/math.2026859

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Research article

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

1.
Mathematics Department, Faculty of Science, Applied Science Private University, Amman, Jordan
2.
Department of Basic Sciences\Scientific, Faculty of Science, Applied Science Private University, Amman, Jordan
3.
Faculty of Mathematics and Data Science, Emirates Aviation University, Dubai, UAE

Received: 22 April 2026 Revised: 23 June 2026 Accepted: 07 July 2026 Published: 16 July 2026
MSC : 26A33, 34A08, 34G10, 46M05, 74D05

We established existence and uniqueness of finite-rank solutions to the conformable fractional abstract Cauchy problem of arbitrary order $ n\alpha $ ($ n \geq 2 $, $ 0 < \alpha \leq 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 \neq 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.
- abstract Cauchy problem,
- conformable fractional derivative,
- Higher-order,
- tensor product of Banach spaces,
- finite-rank function,
- viscoelastic oscillator
Citation: Huda Odetallah, Mayada Abualhomos, Husam Miqdad, Tala Sasa, Lubaba Shaikh, Raja'a Al-Naimi. Finite rank solution for conformable higher-order abstract Cauchy problem in Hilbert space[J]. AIMS Mathematics, 2026, 11(7): 21153-21166. doi: 10.3934/math.2026859

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Abstract

We established existence and uniqueness of finite-rank solutions to the conformable fractional abstract Cauchy problem of arbitrary order $ n\alpha $ ($ n \geq 2 $, $ 0 < \alpha \leq 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 \neq 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.

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