Fractional ODEs and PDEs in the Range (4,5): Theoretical Analysis and Computational Aspects

This paper investigates two nonlinear fractional differential problems where the order of differentiation lies between four and five. The first part focuses on a class of fractional differential equations involving Caputo derivatives of order ? ? (4,5]. Using the Banach contraction principle, we establish the existence and uniqueness of solutions and subsequently analyze their Ulam Hyers stability. Illustrative examples are provided to show the applicability of the two main results. The second part explores the impact of fractional derivatives in wave dynamics by studying traveling wave solutions for the time and space conformable fractional Kawahara equation, where the order is less than five. The tanh method is applied to derive wave solutions, and graphical representations illustrate their behavior. Although treated separately, these two problems share a common feature: both involve fractional differential equations of intermediate order. This connection underscores the significance of such derivatives in both theoretical analysis and applied mathematical modeling.