Fixed Point Method for Nonlinear Fractional Differential Equations with Integral Boundary Conditions on Tetramethyl-Butane Graph

Abstract

Until now, little investigation has been done to examine the existence and uniqueness of solutions for fractional differential equations on star graphs. In the published articles on the subject, the authors used a star graph with one junction node that has edges with the other nodes, although there are no edges between them. These graph structures do not cover more generic non-star graph structures; they are specific examples. The purpose of this study is to prove the existence and uniqueness of solutions to a new family of fractional boundary value problems on the tetramethylbutane graph that have more than one junction node after presenting a labeling mechanism for graph vertices. The chemical compound tetramethylbutane has a highly symmetrical structure, due to which it has a very high melting point and a short liquid range; in fact, it is the smallest saturated acyclic hydrocarbon that appears as a solid at a room temperature of 25 °C. With vertices designated by 0 or 1, we propose a fractional-order differential equation on each edge of tetramethylbutane graph. Employing the fixed-point theorems of Schaefer and Banach, we demonstrate the existence and uniqueness of solutions for the suggested fractional differential equation satisfying the integral boundary conditions. In addition, we examine the stability of the system. Lastly, we present examples that illustrate our findings.

Keywords

• fractional differential equation
• tetramethylbutane graph
• Banach fixed-point theorem
• Caputo derivative
• stability