Conventional Functional Analysis Research Articles

On the Generalized Liouville–Caputo Type Fractional Differential Equations Supplemented with Katugampola Integral Boundary Conditions

In this study, we examine the existence and Hyers–Ulam stability of a coupled system of generalized Liouville–Caputo fractional order differential equations with integral boundary conditions and a connection to Katugampola integrals. In the first and third theorems, the Leray–Schauder alternative and Krasnoselskii’s fixed point theorem are used to demonstrate the existence of a solution. The Banach fixed point theorem’s concept of contraction mapping is used in the second theorem to emphasise the analysis of uniqueness, and the results for Hyers–Ulam stability are established in the next theorem. We establish the stability of Ulam–Hyers using conventional functional analysis. Finally, examples are used to support the results. When a generalized Liouville–Caputo (ρ) parameter is modified, asymmetric results are obtained. This study presents novel results that significantly contribute to the literature on this topic.

Vortex flow, a couple important theorems, and an introduction to distributions

We study vortex flow and perform a number of textbook calculations to highlight inconsistencies between results when using familiar, and widely important, mathematical theorems; these include Stokes’ theorem and a theorem that often enables interchanging field notions like irrotational and conservative, and their connection to scalar potentials and their associated exact differentials. The purpose is to use the familiar setting of a fluid vortex to go beyond conventional functional analysis and introduce students to the theory of distributions. After briefly doing this, we revisit vortex flow using a distributional approach to show how all prior inconsistencies disappear, emphasizing the significance of distributions in physics.