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The widespread use of computer hardware and software in society has led to the emergence of a type of criminal conduct known as cybercrime, which has become a major worldwide concern in the 21st century spanning multiple domains. As a result, in the present setting, academics and practitioners are showing a great deal of interest in conducting research on cybercrime. In this work, a fractional-order model was replaced by involving three sorts of human populations: online computer users, hackers, and cyber security professionals, in order to examine the online computer user-hacker system. The existence, uniqueness and boundedness were studied. To support our theoretical conclusions, a numerical analysis of the influence of the various logical parameters was conducted and we derived the necessary conditions for the different equilibrium points to be locally stable. We examined the effects of the fear level and refuge factor on the equilibrium densities of prey and predators in order to explore and understand the dynamics of the system in a better way. Using some special circumstances, the model was examined. Our theoretical findings and logical parameters were validated through a numerical analysis utilizing the generalized Adams-Bashforth-Moulton technique. [ABSTRACT FROM AUTHOR]

Rumor spreading is the circulation of doubtful messages on the social network. Fact retrieving process that aims at preventing the spread of the rumor, appears to have a significant global impact. In this research, we have investigated a mathematical model projecting rumor spread by considering six groups of individuals namely ignorant, exposed, intentional rumor spreader, unintentional rumor spreader, stifler, and fact retriever. To represent the current abnormal and fast pattern of the message spread around various platforms, in the projected model, we have implemented the fractional derivative in the Caputo context. Using the existing theory of the fractional derivative, we have examined the theoretical aspects such as the existence and uniqueness of the solutions, the existence and stability of the rumor-free and rumor equilibrium points, and the global stability of the rumor-free equilibrium point. Computing basic reproduction numbers, we have analyzed the existence and stability of points of equilibrium. The sensitivity of basic reproduction numbers is also examined. Importance of the fact retrieving drive is highlighted by relating it to the basic reproduction number. Finally, by applying the Adams-Bashforth-Moulton method, we have presented the numerical results by capturing the profile of each of the groups under the influence of fractional derivative and investigated the impact of rumor verification rate and contact rate in controlling and preventing the rumor. With the Caputo fractional operator in the projected model, the current research highlights the significance of the fact retriever and the curb in individual contact and captures the relevant consequences. [ABSTRACT FROM AUTHOR]

In the marine ecosystem, phytoplankton plays a vital role as the primary supplier of oxygen, contributing to 50% of the total oxygen production. Not only does it serve as a significant source of food for other species, but it also sustains life in the ocean. However, the rising ocean temperatures caused by global warming have severely hindered the ability of phytoplankton to generate oxygen. Furthermore, fishes are crucial consumers of oxygen within the marine ecosystem. This research paper presents a model that intricately connects the dynamics of oxygen, phytoplankton, zooplankton, and fish using the Caputo fractional derivative. The model aims to examine the impact of global warming on the collective dynamics by considering the relationship between the rate of oxygen generation, temperature, and time. The paper establishes the existence and uniqueness of solutions and also analyzes the stability of equilibrium points. Numerical simulations are conducted to demonstrate the impact of fractional derivatives and global warming on oxygen depletion and species extinction. [ABSTRACT FROM AUTHOR]

In this paper, we have derived a new operational matrix of fractional integration by using the independence polynomial of a complete bipartite graph and applied it to solve the Poisson equation with Dirichlet boundary conditions. While deriving the operational matrix, the Caputo sense fractional derivatives are considered. Series solutions are found by using the collocation matrix method. The main characteristic of the approach is that it reduces a complex fractional differential equation to a system of algebraic equations. The error bound and computational complexity of the projected algorithm are also investigated. Solutions obtained for the Poisson equation have established the relevance and applicability of the method described. Comparative analysis of the obtained results with the exact solutions convinces that the present method can be considered an efficient numerical tool for solving fractional differential equations. [ABSTRACT FROM AUTHOR]

The Korteweg{De Vries (KdV) equation has always provided a venue to study and generalizes diverse physical phenomena. The pivotal aim of the study is to analyze the behaviors of forced KdV equation describing the free surface criti- cal ow over a hole by finding the solution with the help of q-homotopy analysis transform technique (q-HATT). he projected method is elegant amalgamations of q-homotopy analysis scheme and Laplace transform. Three fractional oper-ators are hired in the present study to show their essence in generalizing the models associated with power-law distribution, kernel singular, non-local and non-singular. The fixed-point theorem employed to present the existence and uniqueness for the hired arbitrary-order model and convergence for the solution is derived with Banach space. The projected scheme springs the series solution rapidly towards convergence and it can guarantee the convergence associated with the homotopy parameter. Moreover, for diverse fractional order the physi- cal nature have been captured in plots. The achieved consequences illuminates, the hired solution procedure is reliable and highly methodical in investigating the behaviours of the nonlinear models of both integer and fractional order. [ABSTRACT FROM AUTHOR]

In this study, we depicted the spread of pests in tea plants and their control by biological enemies in the frame of a fractional-order model, and its dynamics are surveyed in terms of boundedness, uniqueness, and the existence of the solutions. To reduce the harm to the tea plant, a harvesting term is introduced into the equation that estimates the growth of tea leaves. We analyzed various points of equilibrium of the projected model and derived the conditions for the stability of these equilibrium points. The complex nature is examined by changing the values of various parameters and fractional derivatives. Numerical computations are conducted to strengthen the theoretical findings. [ABSTRACT FROM AUTHOR]