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Copper nanoparticles are widely used in many sectors and research endeavors owing to their unique properties, including a large surface area, catalytic capabilities, and high thermal and electrical conductivity. The selection of the base fluid for copper nanoparticles should be contingent upon the anticipated application requirements since various fluids exhibit distinct characteristics that could potentially impact the mobility of the nanoparticles. The present investigation analyzes heat transfer phenomena occurring across a radially stretched surface. The research explores the effects of different states of Cu nanoparticles when combined with base fluids, such as water and silicone oil, on the heat transfer process. The momentum and energy equations are transformed into nonlinear ordinary differential equations by applying the similarity transformation. The boundary value problem-fourth-order (BVP4C) method numerically solves the governing ordinary differential equation for the modeled problem. In addition, the influence of various factors such as the slip parameter, solid volume fraction, Eckert number, Prandtl number, and unsteadiness parameter are examined. It has been discovered that blade-shaped nanoparticles transfer heat as quickly as possible via silicone oil and water. However, for platelet-shaped nanoparticles, a minimum heat transfer rate has been noted. The maximum skin friction coefficient is observed in platelet-shaped nanoparticles, while blade-shaped nanoparticles have the lowest skin friction coefficient.

After extensive examination, scholars have determined that many dynamic systems exhibit intricate connections not only with their current and past states but also with the delay function itself. As a result, their focus shifts towards fractional neutral stochastic differential equations, which find applications in diverse fields such as biology, physics, signal processing, economics, and others. The fundamental principles of existence and uniqueness of solutions to differential equations, which guarantee the presence of a solution and its uniqueness for a specified equation, are pivotal in both the mathematical and physical realms. A crucial approach for analyzing complex systems of differential equations is the utilization of the averaging principle, which simplifies problems by approximating existing ones. Applying contraction mapping principles, we present results concerning the concepts of existence and uniqueness for the solutions of fractional neutral stochastic differential equations. Additionally, we present Ulam-type stability and the averaging principle results within the framework of Lp space. This exploration involved the utilization of Jensen’s, Gröenwall-Bellman’s, Hölder’s, Burkholder-Davis-Gundy’s inequalities, and the interval translation technique. Our findings are established within the context of the conformable fractional derivative, and we provide several examples to aid in comprehending the theoretical outcomes. [ABSTRACT FROM AUTHOR]

In this paper, a fractional food chain system consisting of a Holling type II functional response was studied in view of a fractional derivative operator. The considered fractional derivative operator provided nonsingular as well as a nonlocal kernel which was significantly better than other derivative operators. Fractional order modeling of a model was also useful to model the behavior of real systems and in the investigation of dynamical systems. This model depicted the relationship among four types of species: prey, susceptible intermediate predators (IP), infected intermediate predators, and apex predators. One of the significant aspects of this model was the inclusion of Michaelis-Menten type or Holling type II functional response to represent the predator-prey link. A functional response depicted the rate at which the normal predator consumed the prey. The qualitative property and assumptions of the model were discussed in detail. The present work discussed the dynamics and analytical behavior of the food chain model in the context of fractional modeling. This study also examined the existence and uniqueness related analysis of solutions to the food chain system. In addition, the Ulam-Hyers stability approach was also discussed for the model. Moreover, the present work examined the numerical approach for the solution and simulation for the model with the help of graphical presentations. [ABSTRACT FROM AUTHOR]

Hepatitis B is a worldwide viral infection that causes cirrhosis, hepatocellular cancer, the need for liver transplantation, and death. This work proposed a mathematical representation of Hepatitis B Virus (HBV) transmission traits emphasizing the significance of applied mathematics in comprehending how the disease spreads. The work used an updated Atangana-Baleanu fractional difference operator to create a fractional-order model of HBV. The qualitative assessment and wellposedness of the mathematical framework were looked at, and the global stability of equilibrium states as measured by the Volterra-type Lyapunov function was summarized. The exact answer was guaranteed to be unique using the Lipschitz condition. Additionally, there were various analyses of this new type of operator to support the operator’s efficacy. We observe that the explored discrete fractional operators will be χ²-increasing or decreasing in certain domains of the time scale Nj := j, j +1,... by looking at the fundamental characteristics of the proposed discrete fractional operators along with χ-monotonicity descriptions. For numerical simulations, solutions were constructed in the discrete generalized form of the Mittag-Leffler kernel, highlighting the impacts of the illness caused by numerous causes. The order of the fractional derivative had a significant influence on the dynamical process utilized to construct the HBV model. Researchers and policymakers can benefit from the suggested model’s ability to forecast infectious diseases such as HBV and take preventive action. [ABSTRACT FROM AUTHOR]

This study focused on introducing a fresh model of fractional operators incorporating multiple delays, termed fractional integro-differential Langevin equations with multiple delays. Additionally, the research evaluated the relative controllability of this model within finite-dimensional spaces. Employing fixed-point theory yields the desired outcomes, with the controllability assessment facilitated by Schauder's fixed point and the Grammian matrix defined through the Mittag-Leffer matrix function. Validation of the results was conducted through an application. [ABSTRACT FROM AUTHOR]

The aim of this work is to solve a numerical computation of the neutral fractional functional integro-differential equation based on a new approach to the Legendre wavelet method. The concept of fractional derivatives was examined in the sense of Caputo. The properties of the Legendre wavelet and function approximation were employed to determine the approximate solution of a given dynamical system. Moreover, the error estimations and convergence analysis of the truncated Legendre wavelet expansion for the proposed problem were discussed. The validity and applicability of this proposed technique to numerical computation were shown by illustrative examples. Eventually, the results of this technique demonstrate its great effectiveness and reliability. [ABSTRACT FROM AUTHOR]

The main objective of our research was to explore and develop a fractional-order derivative within the predator-prey framework. The framework includes prey refuge and selective nonlinear harvesting, where the harvesting progressively approaches a threshold value as the density of the harvested population advances. For memory effect, a non-integer order derivative is better than an integer-order derivative. The solutions to the fractional framework were shown to be existence, uniqueness, non-negativity, and boundedness. Matignon's condition was used for analysing local stability, and a suitable Lyapunov function provided global stability. While discussing the Hopf bifurcation's existence condition, we explored derivative order and refuge as bifurcation parameters. We aimed at redefining the predator-prey framework to incorporate fractional order, refuge, and harvesting. This kind of nonlinear harvesting is more realistic and reasonable than the model with constant yield harvesting and constant effort harvesting. The Adams-Bashforth-Moulton PECE algorithm in MATLAB software was used to simulate the proposed outcomes, investigate the impact on various factors, and analyse harvesting's effect on non-integer order predator-prey interactions. [ABSTRACT FROM AUTHOR]

One kind of stochastic delay differential equation in which the delay term is dependent on a proportion of the current time is the pantograph stochastic differential equation. Electric current collection, nonlinear dynamics, quantum mechanics, and electrodynamics are among the phenomena modeled using this equation. A key idea in physics and mathematics is the well-posedness of a differential equation, which guarantees that the solution to the problem exists and is a unique and meaningful solution that relies continuously on the initial condition and the value of the fractional derivative. Ulam-Hyers stability is a property of equations that states that if a function is approximately satisfying the equation, then there exists an exact solution that is close to the function. Inspired by these findings, in this research work, we established the Ulam-Hyers stability and well-posedness of solutions of pantograph fractional stochastic differential equations (PFSDEs) in the framework of conformable derivatives. In addition, we provided examples to analyze the theoretical results. [ABSTRACT FROM AUTHOR]

In numerous domains, fractional stochastic delay differential equations are used to model various physical phenomena, and the study of well-posedness ensures that the mathematical models accurately represent physical systems, allowing for meaningful predictions and analysis. A fractional stochastic differential equation is considered well-posed if its solution satisfies the existence, uniqueness, and continuous dependency properties. We established the well-posedness and regularity of solutions of conformable fractional stochastic delay differential equations (CFrSDDEs) of order γ ∈ (1/2, 1) in Lp spaces with p ≥ 2, whose coefficients satisfied a standard Lipschitz condition. More specifically, we first demonstrated the existence and uniqueness of solutions; after that, we demonstrated the continuous dependency of solutions on both the initial values and fractional exponent γ. The second section was devoted to examining the regularity of time. As a result, we found that, for each π ∈ (0, γ-1/2), the solution to the considered problem has a F-Hölder continuous version. Lastly, two examples that highlighted our findings were provided. The two main elements of the proof were the Burkholder-Davis-Gundy inequality and the weighted norm. [ABSTRACT FROM AUTHOR]