Search

In the present paper, we generalized some of the operators defined in (p , q) -calculus with respect to another function. More precisely, the generalized (p , q) - ϕ -derivatives and (p , q) - ϕ -integrals were introduced with respect to the strictly increasing function ϕ with the help of different orders of the q-shifting, p-shifting, and (q / p) -shifting operators. Then, after proving some related properties, and as an application, we considered a generalized (p , q) - ϕ -difference problem and studied the existence property for its unique solutions with the help of the Banach contraction mapping principle. [ABSTRACT FROM AUTHOR]

In this study, changes in westerly waves and their connections to increased global warming under the influence of greenhouse gases were investigated via a Caputo fractional four-dimensional atmospheric system. The idea of the existence of chaotic behavior in the westerly wind's motion was depicted. It has been noted that westerlies are becoming stronger due to rising air temperatures. An analysis of the existence, uniqueness, boundedness, stability of equilibrium points, and conservative behavior of the solutions was conducted. To prove the existence of chaos in the modified model, the Lyapunov exponents, Poincaré map, and bifurcation were computed. A sliding mode controller to control the chaos in this novel fractional-order system was designed, and conditions for the global stability of the controlled system with and without external disturbances and uncertainties were derived. The finite-time interval for the system to reach the sliding surface was computed. The developed controller's performance was evaluated with respect to both commensurate and non-commensurate fractional derivatives. In each scenario, the impact of fractional orders was investigated. Numerical simulations were used to support theoretical statements about how the controller affects the system. [ABSTRACT FROM AUTHOR]

In the present article, the matrix projective synchronization (MPS) and the inverse matrix projective synchronization (IMPS) have been analyzed with fractional‐order chaotic systems with uncertain terms. First, we theoretically discussed both types of synchronizations. To achieve both types of synchronization, several required conditions are derived. We also drew portraits of Lyapunov's exponents and bifurcation for more analysis. Furthermore, time‐delay terms are also considered, giving an elegant application in real‐world problems. Finally, numerical results agreed with the theoretical hypothesis. The numerical outcomes are shown through the graphical presentation. [ABSTRACT FROM AUTHOR]

We aim to investigate the stability property for the certain linear and nonlinear fractional q -difference equations in the Ulam–Hyers and Ulam–Hyers–Rassias sense. To achieve this goal, we prove that three types of the linear q -difference equations of the q -Caputo-like type are Ulam–Hyers stable by using the quantum Laplace transform and quantum Mittag–Leffler function. Moreover, after proving the existence property for a nonlinear Cauchy q -difference initial value problem, we use the same quantum Laplace transform and the q -Gronwall inequality to show that it is generalized Ulam–Hyers–Rassias stable. [ABSTRACT FROM AUTHOR]

In this paper, a three-dimensional mathematical model of cancer, incorporating tumor cells, host normal cells, and effector immune cells, is examined. The chaotic nature of this cancer model is explored within the framework of Caputo fractional calculus. The analysis employs local stability assessment using the Matignon criteria, Lyapunov exponents (LEs) for various fractional orders of derivatives, bifurcation plots reflecting changes in the fractional derivative order, simulation outcomes of a novel chaotic attractor, Kaplan–Yorke dimension, and sensitivity to initial conditions. The results demonstrate that the Caputo fractional cancer model exhibits chaotic behavior for selected parameters. Moreover, the fractional derivative orders significantly influence the extent of chaotic behavior. Specifically, as the fractional derivative order increases from zero to one, the intensity of chaos diminishes, evidenced by a decrease in both the LE and the Kaplan–Yorke dimension. A piecewise modeling approach is applied to the cancer model, incorporating deterministic, stochastic, and power-law behaviors. Three distinct scenarios are developed within this framework, revealing several novel crossover behaviors through simulation. Notably, it is observed that when cancer dynamics initially follow a power-law behavior and subsequently transition into a stochastic process, the disease dynamics can stabilize, suggesting a positive outlook for disease control. Conversely, if the cancer dynamics begin with a deterministic process and then shift to a stochastic process, the system may enter a chaotic state, complicating disease management efforts. [ABSTRACT FROM AUTHOR]

In this research, we introduce a new and generalized family of convex functions, entitled the α -convex functions in the second sense with respect to a parameter and examine their important algebraic properties. Based on this novel convexity concept, we explore a new class of fractional integral inequalities for functions that are twice differentiable. These results are derived from fundamental identities obtained using classical Riemann–Liouville fractional integrals. To validate our findings, we provide 2D and 3D graphs of the main results. Furthermore, as an additional aspect of our study, we explore error estimates for differences of generalized means. [ABSTRACT FROM AUTHOR]