Search

Global warming is a complex problem with far-reaching global implications. One of its notable repercussions is the escalation of CO2levels in the atmosphere, resulting in the phenomenon known as Ocean Acidification (OA). In this research, we have established a correlation between four key factors: marine species, human population, CO2levels, and ocean pH. By formulating a Caputo fractional differential equation, we investigated the dynamics of these variables to evaluate the significance of this climatic phenomenon. The model analysis reveals that the rise in anthropogenic CO2emissions causes a reduction in the ocean’s pH level and increases OA. This process, in turn, decreases the ocean’s ability to absorb CO2, making it less effective in mitigating climate change. In this study, it was demonstrated that elevated levels of CO2result in a reduction in pH levels, which in turn causes a decrease in the population of marine species that play a critical role in numerous economic sectors such as tourism, aquaculture, and fisheries. Moreover, we conducted a comprehensive analysis of the influence exerted by the intrinsic growth rate of the human population. We examined various theoretical aspects, including the assessment of existence and uniqueness. Numerical simulations are carried out to illustrate the effect of key parameters on the dynamics of the system using the generalized Adams–Bashforth–Moulton method.

The vast applicability of fractional calculus to model physical phenomena in the form of fractional differential equations and their complexity has created a massive demand for efficient analytic and semi-analytic techniques to solve fractional differential equations. This paper has derived a new operational matrix using the independence polynomial of a complete bipartite graph to solve multi-order fractional differential equations. While deriving the operational matrix, the Caputo sense fractional derivatives have been considered. Series solutions are found by using the collocation matrix method. The main characteristic of this approach is that it reduces a complex fractional differential equation to a system of algebraic equations. The convergence analysis and the time complexity analysis of the proposed scheme are also presented in this paper. Six examples have been considered to illustrate the relevance and applicability of the method described. The results obtained are compared with the exact solutions. We have also compared our results with the ones obtained by other methods available in the literature.

In this article, we investigate the dynamics of a three-dimensional class of systems using Caputo fractional derivative. To regulate the chaos of this fractional order system, we propose a novel approach involving a sliding mode controller and an adaptive sliding mode controller. The study includes the existence and Ulam–Hyers stability results, the derivation of conditions ensuring global stability for the controlled system, even in the presence of novel uncertainties and outer interruptions. Our main contribution lies in introducing the adaptive law combined with sliding mode control, which effectively illustrates the chaotic behavior and dissipative nature of the chaotic system. This innovation has promising implications for various real-world applications where the control of chaotic dynamics is critical, such as in secure communication systems or ecological modeling Fitting the Nose–Hoover chaotic system to the defined class of equations, we have analyzed the impact of fractional order derivatives in the system and differentiate between chaotic and hyper chaotic behaviors by employing Lyapunov exponents. This differentiation has significant implications for understanding the underlying dynamics of complex systems in science and engineering. Furthermore, we have performed the bifurcation analysis to depict the chaotic behavior in the novel Nose–Hoover class, shedding light on the rich and intricate behaviors that fractional order dynamics can exhibit. To validate the proposed control scheme, we conduct numerical simulations and demonstrate its efficiency through the imposition of exponential and logarithmic uncertainties and disturbances. The model simulation results are provided.

This study aims to propose a mathematical model in the frame of fractional derivative, that explores the relationship between crops, farmers, and intermediaries. To model this dynamics of the interaction between crops, farmers, and intermediaries, we employ the three-species Lotka–Volterra Holling type-I model. Existence and uniqueness of the solution are examined by defining Picard’s operator and Banach fixed point theorem. We have derived sufficient conditions for the local and global stability of the equilibrium points. Additionally, we conduct numerical simulations to demonstrate the effects of key parameters on the dynamics of the state variables of the system using the generalized Adams–Bashforth–Moulton method. Collecting the data from authorized sources, we have validated the arguments made in the present study.