This paper investigates the existence and uniqueness of solutions for two types of conformable fractional boundary value problems: a classical CFBVP of order $\alpha \in (\frac{3}{2},2]$, and a sequential conformable fractional boundary value problem (SCFBVP) of order $\beta \in (\frac{1}{2},1]$. By establishing new integral bounds for the Green's functions associated with both problems, we extend the results obtained by Z. Laadjal et al. (Numerical Methods for Partial Differential Equations, 40 (2024), e22760) by applying Rus's fixed point theorems. Furthermore, we establish an existence and uniqueness theorem for the SCFBVP based on the Banach fixed point theorem, which complements their findings. Our results improve their work by relaxing key assumptions and broadening applicability. Finally, we present a detailed numerical comparison between the results herein and the existing results, highlighting the advantages of our approach, followed by concluding remarks.

The present study explores the application of hypergeometric functions in evaluating fractional integrals, providing a comprehensive framework to bridge fractional calculus and special functions. As a generalization of classical integrals, fractional integrals have gained prominence due to their wide applicability in modeling anomalous diffusion, viscoelastic systems, and other non-local phenomena. Hypergeometric functions, renowned for their rich analytical properties and ability to represent solutions to differential equations, offer an elegant and versatile tool for solving fractional integrals. In this paper, we evaluate a new class of fractional integrals, presenting results that contribute significantly to the study of generalized hypergeometric functions, particularly ${}_3{F}_2(1)$. The results reveal previously unexplored connections within these functions, providing new insights and extending their applicability. Furthermore, evaluating these fractional integrals holds promise for advancing the theoretical understanding and practical applications of fractional differential equations.