Search Results (19)

The purpose of this paper is to propose a fractal–fractional-order for computer virus propagation dynamics, in accordance with the Atangana–Baleanu operator. We examine the existence of solutions, as well as the Hyers–Ulam stability, uniqueness, non-negativity, positivity, and boundedness based on the fractal–fractional sense. Hyers–Ulam stability is significant because it ensures that small deviations in the initial conditions of the system do not lead to large deviations in the solution. This implies that the proposed model is robust and reliable for predicting the behavior of virus propagation. By establishing this type of stability, we can confidently apply the model to real-world scenarios where exact initial conditions are often difficult to determine. Based on the equivalent integral of the model, a qualitative analysis is conducted by means of an iterative convergence sequence using fixed-point analysis. We then apply a numerical scheme to a case study that will allow the fractal–fractional model to be numerically described. Both analytical and simulation results appear to be in agreement. The numerical scheme not only validates the theoretical findings, but also provides a practical framework for predicting virus spread in digital networks. This approach enables researchers to assess the impact of different parameters on virus dynamics, offering insights into effective control strategies. Consequently, the model can be adapted to real-world scenarios, helping improve cybersecurity measures and mitigate the risks associated with computer virus outbreaks. Full article

We propose the definitions of some fractional integral operators called k-Atangana–Baleanu fractional integral operators. These newly proposed operators are generalizations of the well-known Atangana–Baleanu fractional integral operators. As an application, we establish a generalization of the Hermite–Hadamard inequality. Additionally, we establish some new identities involving these new integral operators and obtained new fractional integral inequalities of the midpoint and trapezoidal type for functions whose derivatives are bounded or convex. Full article

The theory of integral inequality has gained considerable attention due to its influential impact on several fields of mathematics and applied sciences. Over the years, numerous refinements, generalizations, and extensions of convexity have been explored to achieve more precise variants of already established results. The principal idea of this article is to establish some interval-valued integral inequalities of the Hermite–Hadamard type in the fractional domain. First, we propose the idea of generalized interval-valued convexity with respect to the continuous monotonic functions ⋎, bifunction $\zeta$, and based on the containment ordering relation, which is termed as $(⋎,\mathfrak{h})$ pre-invex functions. This class is innovative due to its generic characteristics. We generate numerous known and new classes of convexity by considering various values for ⋎ and $\mathfrak{h}$. Moreover, we use the notion of $(⋎,\mathfrak{h})$-pre-invexity and Atangana–Baleanu (AB) fractional operators to develop some fresh fractional variants of the Hermite–Hadamard (HH), Pachpatte, and Hermite–Hadamard–Fejer (HHF) types of inequalities. The outcomes obtained here are the most unified forms of existing results. We provide several specific cases, as well as a numerical and graphical study, to show the significance of the major results. Full article

In this article, we study the fractional SIR epidemic model with the Atangana–Baleanu–Caputo fractional operator. We explore the properties and applicability of the ZZ transformation on the Atangana–Baleanu–Caputo fractional operator as the ZZ transform of the Atangana–Baleanu–Caputo fractional derivative. This study is an application of two power methods. We obtain a special solution with the homotopy perturbation method (HPM) combined with the ZZ transformation scheme; then we present the problem and study the existence of the solution, and also we apply this new method to solving the fractional SIR epidemic with the ABC operator. The solutions show up as infinite series. The behavior of the numerical solutions of this model, represented by series of the evolution in the time fractional epidemic, is compared with the Adomian decomposition method and the Laplace–Adomian decomposition method. The results showed an increase in the number of immunized persons compared to the results obtained via those two methods. Full article

This manuscript contains several fixed point results for ${\alpha }_{\Gamma }$-F-fuzzy contractive mappings in the framework of orthogonal fuzzy metric spaces. The symmetric property guarantees that the distance function is consistent and does not favour any one direction in orthogonal fuzzy metric spaces. No matter how the points are arranged, it enables a fair assessment of the separations between all of them. In fixed point results, the symmetry condition is preserved for several types of contractive self-mappings. Moreover, we provide several non-trivial examples to show the validity of our main results. Furthermore, we solve non-linear fractional differential equations, the Atangana–Baleanu fractional integral operator and Fredholm integral equations by utilizing our main results. Full article

Salt-and-pepper noise (SPN) is a common type of image noise that appears as randomly distributed white and black pixels in an image. It is also known as impulse noise or random noise. This paper aims to introduce a new weighted average based on the Atangana–Baleanu fractional integral operator, which is a well-known idea in fractional calculus. Our proposed method also incorporates the concept of symmetry in the window mask structures, resulting in efficient and easily implementable filters for real-time applications. The distinguishing point of these techniques compared to similar methods is that we employ a novel idea for calculating the mean of regular pixels rather than the existing used mean formula along with the median. An iterative procedure has also been provided to integrate the power of removing high-density noise. Moreover, we will explore the different approaches to image denoising and their effectiveness in removing noise from images. The symmetrical structure of this tool will help in the ease and efficiency of these techniques. The outputs are compared in terms of peak signal-to-noise ratio, the mean-square error and structural similarity values. It was found that our proposed methodologies outperform some well-known compared methods. Moreover, they boast several advantages over alternative denoising techniques, including computational efficiency, the ability to eliminate noise while preserving image features, and real-time applicability. Full article

The aim of this research is to explore fractional integral inequalities that involve interval-valued preinvex functions. Initially, a new set of fractional operators is introduced that uses the extended generalized Mittag-Leffler function $E_{\mu ,\alpha ,l}^{\gamma ,\delta ,k,c}(\tau ;p)$ as a kernel in the interval domain. Additionally, a new form of Atangana–Baleanu operator is defined using the same kernel, which unifies multiple existing integral operators. By varying the parameters in $E_{\mu ,\alpha ,l}^{\gamma ,\delta ,k,c}(\tau ;p)$, several new fractional operators are obtained. This study then utilizes the generalized AB integral operators and the preinvex interval-valued property of functions to establish new Hermite–Hadamard, Pachapatte, and Hermite–Hadamard–Fejer inequalities. The results are supported by numerical examples, graphical illustrations, and special cases. Full article

This research focuses on the analysis of the competitive model used in the banking sector based on the stochastic fractional differential equation. For the approximate solution, a pseudospectral technique is utilized for the proposed model based on the stochastic Lotka–Volterra equation using a wide range of fractional order parameters in simulations. Conditions for stable and unstable equilibrium points are provided using the Jacobian. The Lotka–Volterra equation is unstable in the long term and can produce highly fluctuating dynamics, which is also one of the reasons that this equation is used to model the problems arising in finance, where fluctuations are important. For this reason, the conventional analytical and numerical methods are not the best choices. To overcome this difficulty, an automatic procedure is used to solve the resultant algebraic equation after the discretization of the operator. In order to fully use the properties of orthogonal polynomials, the proposed scheme is applied to the equivalent integral form of stochastic fractional differential equations under consideration. This also helps in the analysis of fractional differential equations, which mostly fall in the framework of their integrated form. We demonstrate that this fractional approach may be considered as the best tool to model such real-world data situations with very reasonable accuracy. Our numerical simulations further demonstrate that the use of the fractional Atangana–Baleanu operator approach produces results that are more precise and flexible, allowing individuals or companies to use it with confidence to model such real-world situations. It is shown that our numerical simulation results have a very good agreement with the real data, further showing the efficiency and effectiveness of our numerical scheme for the proposed model. Full article

The small size and clever design of nanoparticles can result in large surface areas. This gives nanoparticles enhanced properties such as greater sensitivity, strength, surface area, responsiveness, and stability. This research delves into the phenomenon of a nanobeam vibrating under the influence of a time-varying heat flow. The nanobeam is hypothesized to have material properties that vary throughout its thickness according to a unique exponential distribution law based on the volume fractions of metal and ceramic components. The top of the FG nanobeam is made entirely of ceramic, while the bottom is made of metal. To address this issue, we employ a nonlocal modified thermoelasticity theory based on a Moore–Gibson–Thompson (MGT) thermoelastic framework. By combining the Euler–Bernoulli beam idea with nonlocal Eringen’s theory, the fundamental equations that govern the proposed model have been constructed based on the extended variation principle. The fractional integral form, utilizing Atangana–Baleanu fractional operators, is also used to formulate the heat transfer equation in the suggested model. The strength of a thermoelastic nanobeam is improved by performing detailed parametric studies to determine the effect of many physical factors, such as the fractional order, the small-scale parameter, the volume fraction indicator, and the periodic frequency of the heat flow. Full article

Fractional integrals and inequalities have recently become quite popular and have been the prime consideration for many studies. The results of many different types of inequalities have been studied by launching innovative analytical techniques and applications. These Hermite–Hadamard inequalities are discovered in this study using Atangana–Baleanu integral operators, which provide both practical and powerful results. In this paper, a symmetric study of integral inequalities of Hermite–Hadamard type is provided based on an identity proved for Atangana–Baleanu integral operators and using functions whose absolute value of the second derivative is harmonic convex. The proven Hermite–Hadamard-type inequalities have been observed to be valid for a choice of any harmonic convex function with the help of examples. Moreover, fractional inequalities and their solutions are applied in many symmetrical domains. Full article

We prove a new Taylor’s theorem for generalized weighted fractional calculus with nonsingular kernels. The proof is based on the establishment of new relations for nth-weighted generalized fractional integrals and derivatives. As an application, new mean value theorems for generalized weighted fractional operators are obtained. Direct corollaries allow one to obtain the recent Taylor’s and mean value theorems for Caputo–Fabrizio, Atangana–Baleanu–Caputo (ABC) and weighted ABC derivatives. Full article

The Atangana–Baleanu fractional integral and multiplier transformations are two functions successfully used separately in many recently published studies. They were previously combined and the resulting function was applied for obtaining interesting new results concerning the theories of differential subordination and fuzzy differential subordination. In the present investigation, a new approach is taken by using the operator previously introduced by applying the Atangana–Baleanu fractional integral to a multiplier transformation for introducing a new subclass of analytic functions. Using the methods familiar to geometric function theory, certain geometrical properties of the newly introduced class are obtained such as coefficient estimates, distortion theorems, closure theorems, neighborhoods and the radii of starlikeness, convexity, and close-to-convexity of functions belonging to the class. This class may have symmetric or assymetric properties. The results could prove interesting for future studies due to the new applications of the operator and because the univalence properties of the new subclass of functions could inspire further investigations having it as the main focus. Full article

Applications of the Atangana–Baleanu fractional integral were considered in recent studies related to geometric function theory to obtain interesting differential subordinations. Additionally, the multiplier transformation was used in many studies, providing elegant results. In this paper, a new operator is defined by combining those two prolific functions. The newly defined operator is applied for introducing a new subclass of analytic functions, which is investigated concerning certain properties, such as coefficient estimates, distortion theorems, closure theorems, neighborhoods and radii of starlikeness, convexity and close-to-convexity. This class may have symmetric or asymmetric properties. The results could prove interesting due to the new applications of the Atangana–Baleanu fractional integral and of the multiplier transformation. Additionally, the univalence properties of the new subclass of functions could inspire researchers to conduct further investigations related to this newly defined class. Full article

We propose, in the present paper, to derive some differential subordination results. The work is developed in the case of analytic functions defined on the open unit disc. The results will be formulated by making use of an Atangana–Baleanu fractional integral operator and Bessel functions. For the newly obtained theorems, certain interesting consequences are also considered. Univalent function selections with specific symmetry properties were involved. Full article

In this article, first, we deduce an equality involving the Atangana–Baleanu ($\mathbf{AB}$)-fractional integral operator. Next, employing this equality, we present some novel generalization of Ostrowski type inequality using the Hölder inequality, the power-mean inequality, Young’s inequality, and the Jensen integral inequality for the convexity of $\left|\mathsf{{\rm Y}}\right|$. We also deduced some new special cases from the main results. There exists a solid connection between fractional operators and convexity because of their fascinating properties in the mathematical sciences. Scientific inequalities of this nature and, particularly, the methods included have applications in different fields in which symmetry plays a notable role. It is assumed that the results presented in this article will show new directions in the field of fractional calculus. Full article