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The main goals of this paper are to provide an introduction to the idea of interval-valued n -polynomial s -type convex functions and to investigate the algebraic properties of this type of function. This new generalization aims to show the existence of new Hermite–Hadamard inequalities for the recently presented class of interval-valued n -polynomials of s -type convex describing the φ -fractional integral operator. In the classical sense, some special cases are figured out, and the two examples are also given. There are some recently discovered inequalities for interval-valued functions that are regulated by fractional calculus applicable to interval-valued n -polynomial s -type convexity. The results obtained show that future research will be simple to implement, highly efficient, feasible, and extremely precise in its investigation. It could also help solve modeling problems, optimization problems, and fuzzy interval-valued functions that involve both discrete and continuous variables. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to introduce a more generalized class of convex stochastic processes and explore some of their algebraic properties. This new class of stochastic processes is called the n -polynomial s -type convex stochastic process. We demonstrate that this new class of stochastic processes leads to the discovery of novel Hermite–Hadamard type inequalities. These inequalities provide upper bounds on the integral of a convex function over an interval in terms of the moments of the stochastic process and the convexity parameter s. To compare the effectiveness of the newly discovered Hermite–Hadamard type inequalities, we also consider other commonly used integral inequalities, such as Hölder, Hölder–Ïşcan, and power-mean, as well as improved power-mean integral inequalities. We show that the Hölder–Ïşcan and improved power-mean integral inequalities provide a better approach for the n -polynomial s -type convex stochastic process than the other integral inequalities. Finally, we provide some applications of the Hermite–Hadamard type inequalities to special means of real numbers. Our findings provide a useful tool for the analysis of stochastic processes in various fields, including finance, economics, and engineering. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of numerical solution and stability of a chemostat model under fractal-fractional order derivative. First, we investigate the positivity and roundedness of the solution of the considered system. Second, we find the existence of a solution of the considered system by employing the Banach and Schauder fixed-point theorems. Furthermore, we obtain a sufficient condition that allows the existence of the stabling of solutions by using the numerical-functional analysis. We find that the proposed system exists as a unique positive solution that obeys the criteria of Ulam–Hyers (U-H) and generalized U-H stability. We also establish a numerical analysis for the proposed system by using a numerical scheme based on the Lagrange interpolation procedure. Finally, we provide two numerical examples to verify the correctness of the theoretical results. We remark that the structure described by the considered model is also sometimes called side capacity or cross-flow model. The structure considered here can be also seen as a limiting case of the pattern chemostats in parallel with diffusion connection. Moreover, the said model forms in natural and engineered systems and can significantly affect the hydrodynamics in porous media. Fractal calculus is an excellent tool to discuss fractal characteristics of porous media and the characteristic method of the porous media. [ABSTRACT FROM AUTHOR]

These days the whole world is facing a serious problem of infectious maladies and how to control the endemic of these diseases. Testing correctly is one of the most important procedures in preventing the spread of infectious diseases, as incorrect testing can turn a susceptible person into an infected person. In this paper, we study the dynamics of imperfect testing and diagnostics of infectious diseases model by replacing the integer order derivative in the sense of fractional order Atangana–Baleanu operator coupled with Caputo operator. This AB-fractional operator is the generalization of classical derivative and gives more data of the factors of the nonlocal dynamical frameworks. Fixed point theorems have been used for the verification of existence results and Picard's stability technique utilized for the stability study of a fractional order imperfect testing infectious disease (ITID) model. Finally, numerical computations are implemented for the fractional order ITID model to illustrate the results graphically. [ABSTRACT FROM AUTHOR]

On the grounds of few perceived Gronwall inequalities, we inspect some new relevant nonlinear dynamic inequalities on time scales related to one independent variable. To guarantee the evenness of information division and with the conversation of peculiar cases, it is indicated that these inequalities catch dynamic, delay, Volterra–Fredholm, discrete, fractional, etc. The inequalities suggested right here to other specific bounds on unknown parameters may be handled as powerful equipment for determining the features of various dynamic equations on time scales. It is trusted that this research plan will open new possibilities in in-depth examination of time scale scheme structure of varying nature. [ABSTRACT FROM AUTHOR]

This work is devoted to derive some existence and uniqueness (EU) conditions for the solution to a class of nonlinear delay non-autonomous integro-differential Cauchy evolution problems (CEPs) under Caputo derivative of fractional order. The required results are derived via topological degree method (TDM). TDM is a powerful tool which relaxes strong compact conditions by some weaker ones. Hence, we establish the EU under the situation that the nonlinear function satisfies some appropriate local growth condition as well as of non-compactness measure condition. Furthermore, some results are established for Hyers–Ulam (HU) and generalized HU (GHU) stability. Our results generalize some previous results. At the end, by a pertinent example, the results are verified. [ABSTRACT FROM AUTHOR]

Discrete fractional calculus (DFC) is suggested to interpret neural schemes with memory impacts. This study seeks to formulate some discrete fractional nonlinear inequalities with ξ ̂ fractional sum operators that are used with some conventional and forthright inequalities. Taking into consideration, we recreate the explicit bounds of Gronwall-type inequalities by observing the principle of DFC for unknown functions here. Such inequalities are of new version relative to the current literature findings so far and can be used as a helpful method to evaluate the numerical solutions of discrete fractional differential equations. We show a few uses of the rewarded inequalities to mirror the advantages of our work. Regarding applications, we can apply the acquainted results to discuss boundedness, uniqueness, and continuous dependency on the initial value problem for the solutions of certain underlying worth problems of fractional difference equations. The leading consequences may be proven by the usage of the analysis process and the methodology of the mean value theorem. These variations can be used as an advantageous device in the subjective examination solutions of discrete fractional difference equations. [ABSTRACT FROM AUTHOR]

This paper deals with the global dynamics of deterministic-stochastic COVID-19 mathematical model with quarantine class and incorporating a preventive vaccination. Lyapunov functions are utilized for the global stability of disease free equilibrium point and the graph theoretic method is used for the construction of Lyapunov function for positive equilibrium point. The stability of model is discussed regarding the reproductive number. Utilizing the non-standard finite difference scheme for the numerical solution of the deterministic model, the obtained results are shown graphically. Further, environmental noises are added to the model for description of stochastic model. Then we take out the existence and uniqueness of positive solution with extinction for infection. Finally, we solve numerically the stochastic model using Newton Polynomial scheme and present the results graphically. [ABSTRACT FROM AUTHOR]