Department of Mathematics
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Item Fixed point method for nonlinear fractional differential equations with integral boundary conditions on tetramethyl-butane graph(MDPI, 2024-06) Agarwal, Shivi; Mathur, TrilokUntil 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.Item Fractional-order crime propagation model with non-linear transmission rate(Elsevier, 2023-04) Agarwal, Shivi; Mathur, TrilokVarious studies present different mathematical models of ordinary and fractional differential equations to reduce delinquent behavior and encourage prosocial growth. However, these models do not consider the non-linear transmission rate, which depicts reality better than the linear transmission rate, as the relationship between non-criminals and criminals is not linear. In light of this, a novel fractional-order mathematical crime propagation model with a non-linear Beddington–DeAngelis transmission rate is proposed that divides the entire population into three clusters. The present study also compares the crime transmission models for various transmission rates, followed by an analytical investigation. The model shows two equilibrium points (criminal-free and crime-persistence equilibrium). The criminal-free equilibrium is locally and globally asymptotically stable when the criminal generation number is less than one. The crime-persistence equilibrium point does not appear until the criminal generation number exceeds one. In addition, this research investigates the incidence of transcritical bifurcation at the criminal-free equilibrium point. Furthermore, numerical simulations are performed to demonstrate the analytical results. In summary, the finding of this research suggests that as the order of derivative increases, the population approaches equilibrium more swiftly, and criminals decline with time for the different order of derivative.Item Underlying dynamics of crime transmission with memory(Elsevier, 2021-05) Agarwal, Shivi; Mathur, TrilokVarious studies suggest different mathematical models of integer order differential equations predict crime. But these models do not inherit non-local property, which depicts behavior changes due to contact with criminals for a long period. To overcome this, a fractional-order mathematical model of crime transmission is proposed in this study. The proposed model considers the previous effects of the input while predicting the crime growth rate. A mathematical model of crime transmission inherited with memory property is proposed in this study to analyze crime congestion. Abstract compartmental parameters of fractional crime transmission equation, which illustrates various stages of criminal activity, were employed to analyze crime contagion in the society. The present study demonstrates the progression of the flow of population by classifying into three systems based on involvement in crime and imprisonment by considering the criminal history of an individual. Subsequently, the equilibria of the three-dimensional fractional crime transmission model are evaluated using phase-plane analysis. The Lyapunov function is employed to determine threshold conditions to achieve a crime-free society.