Finite Rank Solution for Conformable Second-Order Abstract Cauchy Problem in Hilbert Space (European Journal of Pure and Applied Mathematics, 2025)
This paper presents a comprehensive analytical framework for constructing finite-rank solution to second-order conformablefractional 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 linearoperators 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.