(untitled) (European Journal of Pure and Applied Mathematics, 2025)
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)=u_0 and u^(α )(0)=u_0^(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.