Search Results (28)

In this paper, we establish new Milne–Mercer-type inequalities via Atangana–Baleanu conformable fractional integral operators for differentiable functions whose derivatives in absolute value are h-convex. First, we derive a novel identity involving the Atangana–Baleanu conformable fractional integral operators. Then, by employing the properties of h-convex functions and fractional integral operators, several new inequalities of the Milne–Mercer type are obtained. The results presented in this paper extend and generalize various previously known inequalities, including classical Milne inequalities, Riemann–Liouville fractional integral inequalities, and conformable fractional integral inequalities. Moreover, several special cases are discussed to demonstrate the generality and applicability of the obtained results. The findings provide new refinements in the theory of fractional integral inequalities and contribute to the development of convex analysis within fractional frameworks. Full article

This paper introduces a multiplicative Atangana–Baleanu–conformable fractional integral operator in the setting of multiplicative calculus. The proposed operator is formulated by applying the Atangana–Baleanu–conformable fractional integral structure to the logarithmic representation of positive functions, thereby combining multiplicative behavior, nonsingular memory effects, and conformable scaling in a single framework. Appropriate function-space assumptions are imposed to ensure that the operator is well defined. Based on this operator, we establish a new auxiliary identity and derive several multiplicative Milne–Mercer-type inequalities for multiplicatively convex functions. The obtained results include multiplicative Riemann–Liouville-type, multiplicative Atangana–Baleanu-type, and conformable-type inequalities as special cases under suitable choices of the parameters. To clarify the role of the fractional parameters, numerical examples are provided together with logarithmic gap values, relative-error comparisons, heatmaps, contour plots, and parameter-sensitivity analyses. These computations illustrate the validity of the derived inequalities and compare the proposed bounds with their reduced special cases. Full article

In this study, we obtain new Hermite–Hadamard type inequalities involving an extended form of the Atangana–Baleanu fractional integral operator having Mittag-Leffler kernels. The approach is based on a suitable integral identity for differentiable functions together with the convexity of the absolute value of the first derivative. Within this framework, we extend the classical Hermite–Hadamard inequality to a fractional setting governed by the parameters $\alpha \in (0,1)$, $\beta \in (0,1]$, and $\lambda >0$. Full article

In this research work, we investigate a nonlinear fractional initial value problem involving the Atangana-Baleanu-Caputo derivative of order $0<\alpha <1.$ By means of the associated fractional integral operator, the problem is converted into an equivalent nonlinear integral equation. The existence of solutions is established in the context of extended $\mathcal{F}$-metric spaces via a fixed point approach based on an ($\alpha ,\psi$)-contractive condition of rational form. Furthermore, we develop the notion of graphic rational contractions in the setting of extended $\mathcal{F}$-metric spaces and prove new fixed point results. Our results extend and unify several known results in the existing literature as special cases. Nontrivial examples are provided to demonstrate the applicability of the theoretical findings. These results highlight the effectiveness of extended $\mathcal{F}$-metric techniques in the analys. Full article

The chain rule is a foundational concept in calculus, critical for differentiating composite functions, especially those appearing in modern AI techniques. Its extension to fractional calculus presents challenges due to the integral-based nature and intrinsic memory effects of these fractional operators. This survey provides a review of chain-rule formulations across major known FDs, including Riemann-Liouville (RL), Caputo, Caputo-Fabrizio (CF), Atangana-Baleanu-Riemann (ABR), Atangana-Baleanu-Caputo (ABC), and Caputo-Fabrizio with Gaussian kernel (CFG). The main contribution here is the introduction of a unified criterion, denoted as $C$, which synthesizes and extends previous classification frameworks for systematically formulating the chain rule across different operators. Each chain rule is examined in terms of its derivation, operator structure, and scope of applicability. In addition, the survey analyzes series-based approximations that appear in computing these derivatives, highlighting the minimum number of terms required to achieve acceptable mean absolute error (MAE). By consolidating theoretical developments, derivation methods, and numerical strategies, this paper provides a comprehensive resource for researchers and practitioners working with fractional-order models. Full article

In this study, we address a coupled system of nonlinear fractional delay differential equations subject to cross-coupled multi-point boundary conditions. By utilizing the generalized power Caputo fractional derivative, we present a unified theoretical framework that encompasses several operators—including the Atangana–Baleanu, Caputo–Fabrizio, and weighted Hattaf derivatives—as special cases. This generality ensures that our results remain applicable across a broad family of fractional kernels. We transform the complex delay system into an equivalent integral form to derive sufficient criteria for the existence and uniqueness of solutions via fixed-point theory. Furthermore, we rigorously establish the Ulam–Hyers stability of the system, a critical property for ensuring robustness in the presence of perturbations. Finally, the theoretical findings are validated through a detailed numerical study employing a predictor–corrector scheme adapted for fractional delay systems. The simulations highlight the sensitivity of solutions to the memory kernel and fractional orders and include a systematic exploration of delay effects. Full article

This paper introduces a fractional generalization of the classical Legendre differential equation based on the Atangana–Baleanu–Caputo (ABC) derivative. A novel fractional Legendre-type operator is rigorously defined within a functional framework of continuously differentiable functions with absolutely continuous derivatives. The associated initial value problem is reformulated as an equivalent Volterra integral equation, and existence and uniqueness of classical solutions are established via the Banach fixed-point theorem, supported by a proved Lipschitz estimate for the ABC derivative. A constructive solution representation is obtained through a Volterra–Neumann series, explicitly revealing the role of Mittag–Leffler functions. We prove that the fractional solutions converge uniformly to the classical Legendre polynomials as the fractional order approaches unity, with a quantitative convergence rate of order $O(1-\alpha )$ under mild regularity assumptions on the Volterra kernel. A fully reproducible quadrature-based numerical scheme is developed, with explicit kernel formulas and implementation algorithms provided in appendices. Numerical experiments for the quadratic Legendre mode confirm the theoretical convergence and illustrate the smooth interpolation between fractional and classical regimes. An application to time-fractional diffusion in spherical coordinates demonstrates that the operator arises naturally in physical models, providing a mathematically consistent tool for extending classical angular analysis to fractional settings with memory. Full article

This research develops a novel coupled system of nonlinear hybrid stochastic fractional differential equations that integrates neutral effects, stochastic perturbations, and hybrid switching mechanisms. The system is formulated using the Atangana–Baleanu–Caputo fractional operator with a non-singular Mittag–Leffler kernel, which enables accurate representation of memory effects without singularities. Unlike existing approaches, which are limited to either neutral or hybrid stochastic structures, the proposed framework unifies both features within a fractional setting, capturing the joint influence of randomness, history, and abrupt transitions in real-world processes. We establish the existence and uniqueness of mild solutions via the Picard approximation method under generalized Carathéodory-type conditions, allowing for non-Lipschitz nonlinearities. In addition, mean-square Mittag–Leffler stability is analyzed to characterize the boundedness and decay properties of solutions under stochastic fluctuations. Several illustrative examples are provided to validate the theoretical findings and demonstrate their applicability. Full article

This work develops a Boundary Element Method (BEM) formulation for simulating bioheat transfer in perfused biological tissues using the Atangana–Baleanu fractional derivative in the Caputo sense (ABC). The ABC operator incorporates a nonsingular Mittag–Leffler kernel to model thermal memory effects while preserving compatibility with standard boundary conditions. The formulation combines boundary discretization with cell-based domain integration to account for volumetric heat sources, and a recursive time-stepping scheme to efficiently evaluate the fractional term. The model is applied to a one-dimensional cylindrical tissue domain subjected to metabolic heating and external energy deposition. Simulations are performed for multiple fractional orders, and the results are compared with classical BEM $(a=1.0)$, Caputo-based fractional BEM, and in vitro experimental temperature data. The fractional order $a\approx 0.894$ yields the best agreement with experimental measurements, reducing the maximum temperature error to 1.2% while maintaining moderate computational cost. These results indicate that the proposed BEM–ABC framework effectively captures nonlocal and time-delayed heat conduction effects in biological tissues and provides an efficient alternative to conventional fractional models for thermal analysis in biomedical applications. Full article

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