Manar. A. Alqudah

In this paper, we investigate a class of biparametrized integral inequalities in the setting of local fractional calculus on fractal sets. Motivated by earlier results in [ 27 , 39 ], we derive a new integral identity associated with a three-point Newton–Cotes formula involving two parameters. A noteworthy feature of this formula is that, through suitable choices of the parameters, it unifies a broad family of one-point, two-point, and three-point quadrature rules. Based on this identity, we establish several new inequalities for functions whose first-order local fractional derivatives satisfy generalized s-convexity assumptions. Our results not only extend a number of existing inequalities in the literature, but also recover several known estimates as particular cases. To illustrate the effectiveness of the theoretical findings, we present an example supported by two- and three-dimensional graphical representations, together with some applications related to quadrature formulas and special means.

Fractional calculus provides an effective framework for modeling memory effects, but classical discrete fractional operators may exhibit structural inconsistencies at the initial node. In this paper, we introduce normalized nabla Atangana–Baleanu and Caputo–Fabrizio fractional operators together with their associated normalized fractional sums. The proposed normalization preserves the nonlocal memory effects while ensuring well-posedness. We establish generalized nabla Grönwall-type inequalities and comparison principles involving discrete Mittag-Leffler kernels, leading to sufficient conditions for the existence, uniqueness, stability, and boundedness of solutions. Explicit formulations are derived for representative normalized fractional initial value problems, including a discrete recurrence formula and an equivalent continuous Volterra integral equation. Numerical results confirm the theoretical analysis and show that both discrete and continuous solutions approach their respective classical limits as the fractional order increases.

In this paper, we study a Caputo–Fabrizio fractional-order climate model to capture the intrinsic nonlocal and fading-memory characteristics of coupled environmental processes. The framework incorporates five interacting state variables surface temperature [Formula: see text], atmospheric carbon concentration [Formula: see text], ice fraction [Formula: see text], atmospheric activity [Formula: see text], and oceanic carbon uptake [Formula: see text]. The non-singular exponential kernel embedded in the Caputo–Fabrizio operator ensures smooth temporal weighting and realistic climate inertia without introducing singular behavior. Theoretical analysis establishes existence, uniqueness, and dynamical stability of the proposed system within a fractional framework. For computational investigation, we perform numerical analysis using a fractional weighted discrete iterative scheme (FWDIS), which preserves the exponential memory structure while enabling stable and efficient time-marching of the nonlinear coupled model. Furthermore, an artificial neural network trained via the Levenberg–Marquardt optimization algorithm is employed to learn complex climate interactions and validate numerical accuracy. Strong agreement between analytical findings, discrete simulations, and intelligent predictions demonstrates that integrating fractional calculus with advanced numerical and learning strategies provides a robust and forward-looking framework for long-term climate forecasting and environmental risk assessment.

In this study, the transmission dynamics of a fractional-order host–vector mathematical model of pine wilt disease is investigated. The Atangana–Baleanu fractional-order derivative is applied instead of classical-order derivative for incorporating non-local and memory effects in the system through fractional parameter. The considered model is rigorously studied for existence and uniqueness, in addition with Ulam–Hyers stability, providing a strong theoretical base for the model. A brief discussion on the disease-free equilibrium and endemic equilibrium points is presented using basic reproduction number [ ] of the system. Furthermore, an efficient numerical method is applied to obtain the approximate solution of the proposed system. The graphical results show that the fractional parameter [ ] has a significant impact on the disease dynamics. When the value of [ ] decreases, then the memory gets stronger, hence past history affects the present transmission dynamics and when the value of [ ] increases ([ ]), the memory becomes weaker and the fractional model tends to its integer-order counterpart. The plots of all variables affirm the significant effect of fractional parameter in the system dynamics. By using the important concept of artificial intelligence-based deep neural networks, some important thresholds like mean squared error and root mean squared error as well as the regression coefficients are analyzed. The absolute error between the numerical results and those predicted by deep neural networks is compared. For the mentioned computational purposes, the Levenberg–Marquardt algorithm and MATLAB 2023 are used. Graphical illustrations with performance analysis for test, validation, train and all data are displayed.

This paper explores the extension of classical fractional operators to the framework of [ ]-calculus, a non-Newtonian calculus in which differentiation and integration are defined via multiplicative analogs of their classical counterparts. We begin by recalling key concepts from both fractional calculus and [ ]-calculus. Next, we revisit the recently introduced multiplicative Riemann–Liouville fractional operators and extend the multiplicative Riemann–Liouville fractional derivative to arbitrary order [ ]. Building on this foundation, we introduce multiplicative versions of the Hadamard and Katugampola fractional integrals and derivatives. Finally, we establish Hermite–Hadamard inequalities for both newly defined integrals.

A class of hybrid fractional differential equations is the subject of the current investigation. Differential operator with Mittag-Leffler-type kernel is used. Proportional delay, a crucial component in electrodynamics and infectious illnesses, is included in the problem. Several factors, including qualitative analyses, Ulam–Hyer-type stability, and a general numerical scheme, have been taken into consideration for the suggested problem. We used functional analysis concepts, in particular the theory of fixed points, to general analysis. Lagrange’s interpolation approach, an existence methodology, has been used to study numerical aspect of the considered model. By using the adopted results, an example is studied at the conclusion to support the proposed theory.

This study investigates the existence and uniqueness of solutions to initial value problems for nonlinear variable-order weighted fractional differential equations with finite delay. Building upon and generalizing prior constant-order fractional models, our approach employs fixed-point theory, specifically the Banach and Schauder fixed-point theorems, in suitable weighted function spaces to rigorously establish these fundamental results. We further demonstrate the applicability of our theoretical framework through illustrative examples. The findings contribute significantly to the mathematical understanding and modeling capabilities of complex systems exhibiting memory and hereditary properties governed by variable-order fractional dynamics.

This paper proposes a collection of nonlinear differential operators. The goal of the work is to develop certain theoretical and analytical intuitions regarding the replacement of the general deferential operators with the derivatives of Caputo, Atangana–Baleanu, and Caputo–Fabrizio. A variety of well-known methodologies were used to examine the models. The Lagrange interpolation polynomial was used to establish a numerical method to solve the models under investigations. We have investigated the related perspective on existence and uniqueness through this work. Several numerical findings were incorporated. The updated models examine the impact of competition on prey and predator abundance. A species’ variations in abundance are modeled as a function of the abundance of its competitors; the specific competitive mechanism is neither investigated nor stated in explicit terms.

The aim of this paper is to study the stability analysis and closed-form solutions of conformable time-fractional Zoomeron equation in porous media. By utilizing function transformation and adjusting indexes of the transformation via Power Index Method (PIM), we can reduce the partial differential equation to ordinary differential equation. The exact solutions of the ordinary differential equation can be obtained by using symbolic package Maple. Our findings are more general and contain fractional derivative parameter. The study shows that the used method is effective and reliable. We apply the conformable derivative and illustrated the application of Zoomeron equation in fractal porous media with traveling wave transformation. We have obtained the trajectory of Zoomeron equation by using eigenvalues and eigenvectors of Jacobian matrices at fixed points.