On the novel existence results of solutions for a class of fractional boundary value problems on the cyclohexane graph

Abstract

A branch of mathematical science known as chemical graph theory investigates the implications of connectedness in chemical networks. A few researchers have looked at the solutions of fractional differential equations using the concept of star graphs. Their decision to use star graphs was based on the assumption that their method requires a common point linked to other nodes but not to each other. Our goal is to broaden the scope of the method by defining the idea of a cyclohexane graph, which is a cycloalkane with the molecular formula $C_{6}H_{12}$ C 6 H 12 and CAS number 110-82-7. It consists of a ring of six carbon atoms, each bonded with two hydrogen atoms above and below the plane with multiple junction nodes. This article examines the existence of fractional boundary value problem’ solutions on such graphs in the sense of the Caputo fractional derivative by using the well-known fixed point theorems. In addition, an example is given to support our key findings.

A Comprehensive Review on Adsorption, Photocatalytic and Chemical Degradation of Dyes and Nitro-Compounds over Different Kinds of Porous and Composite Materials

“…Numerous researchers have worked on different sources for biochar production, such as rice straw, Korean cabbage, wood chip, pine wood biochar, pig manure biochar, paper-derived biochar, peels of oranges, peels of bananas, coir pith, date pith, peanut shells, almond hulls, Citrullus lanatus rind, ZnO/cotton stalk biochar, cow dung biochar, litchi peel biochar, sludge biochar, low-rank coal (leonardite), coffee waste-activated biochar, oil palm waste biochar, waste palm shell biochar, wet-torrefied microalgal biochar, and green biochar/iron oxide, for the adsorption of various dyes (congo red, crystal violet, methylene blue, aid violet, basic blue 9, reactive dye, acid yellow 36, rhodamine B, malachite green, etc.) [ 72 , 73 , 74 , 75 , 76 , 77 , 78 , 79 , 80 , 81 ]. From the thermodynamic parameters as reported by Zhang et al, physical sorption is the primary adsorption mechanism through which methylene blue adsorbs biochar/Fe x O y [ 80 ].…”

Existence, uniqueness, and Ulam stability of solutions of fractional conformable Langevin system on the ethane graph

“…In 2022, Ali et al [17] researched the fractional boundary value problems on cyclohexane graph. They proceeded to investigate the existence of solutions to specific problems in their research: $$$$ \left\{\begin{array}{l}{\mathcal{D}}^{\beta }{u}_w(y)={\mathcal{F}}_w\left(y,{u}_w(y),{\mathcal{D}}^{\alpha }{u}_w(y),{u}_w^{\prime }(y)\right),y\in \left[0,1\right],\\ {}{\delta}_1{u}_w(0)+{\delta}_2{u}_w(1)={\delta}_3{\int}_0^1{\mathcal{D}}^{\beta -1}{u}_w\left(\theta \right) d\theta, \\ {}{\delta}_1{\mathfrak{D}}^{\alpha }{u}_w(1)+{\delta}_2{\mathfrak{D}}^{2\alpha }{u}_w(1)&a...$$…”

“…Based on the above work, fractional differential systems have been extensively studied on different graphs, such as cyclohexane [17], glucose [21], and methylpropane [22]. Ethane (refer to Figure 2) is the second member of the alkane series and is the only aliphatic hydrocarbon in the alkane composed of two carbon atoms.…”

General semi-implicit approximations with errors for common fixed points of nonexpansive-type operators and applications to Stampacchia variational inequality

“…More applications of fixed point theory, such as equations, image process and other optimization problems, one can refer to Cacciapaglia and Sannino ( 2021 ); Ali et al. ( 2022 ); Panda et al. ( 2020 ); Harker and Pang ( 1990 ); Ali et al.…”