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

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:% {equation*} Eu(2α )(t)+Au(α )(t)+Bu(t)=f(t) {equation*} subject to prescribed initial conditions u(0)=u₀ and u(α )(0)=u₀(α ), where $A,$ B and E represent closed linear operators acting on a Banach space $X,$ f:[0,∞ )→ X is continuous, and u is continuously differentiable on [0,∞ ). 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.