Finite Rank Solution for Conformable Second-Order Abstract Cauchy Problem in Hilbert Space

Abstract

This paper presents a comprehensive analytical framework for constructing finite-rank solution to second-order conformable

fractional abstract Cauchy problem. We examine the mathematical structure:%

\begin{equation*}

Eu^{(2\alpha )}(t)+Au^{(\alpha )}(t)+Bu(t)=f(t)

\end{equation*}

subject to prescribed initial conditions $u(0)=u_{0}$ and $u^{(\alpha

)}(0)=u_{0}^{(\alpha )},$ where $A,$ $B$ and $E$ represent closed linear

operators acting on a Banach space $X,$ $f:[0,\infty )\rightarrow X$ is continuous, and $u$ is continuously differentiable on $[0,\infty ).$ Our analytical methodology exploits tensor product decomposition techniques to transform the problem into finite-dimensional systems. This work proves solution existence and uniqueness under specific conditions, and provides computational methods for many types of this problem.