Search Results (169)

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

The analysis of the time-fractional nonlinear Sharma–Tasso–Olver (STO) equation with various initial conditions has been shown in this work. Finding the appropriate approximate solution of the problems under consideration is carried out by implementing unique strategies that combine the Adomian decomposition method (ADM), and the Generalized integral transform. The proposed method computes the results as a convergent series. The main benefit of the suggested method is that it needs minimal computing effort while producing extremely accurate results. We first apply the fractional Caputo fractional derivative (CFD) and then the Atangana–Baleanu–Caputo (ABC) derivative to solve the fractional STO problem. The nonlinear wave model for harbor and coastal designs heavily relies on the wave solutions of the STO equation. Several cases of time-fractional STO equations with various initial approximations are used to illustrate the schemes under consideration. The efficiency and dependability of the methods under consideration are confirmed by executing suitable numerical simulations. We contrast our findings with those of other approaches, including the Homotopy perturbation method (HPM), and the q-Homotopy analysis Elzaki transform method (q-HAETM). Additionally, the results of using the proposed techniques at different fractional orders are analyzed, showing that their accuracy increases as the value goes from fractional order to integer order. The results gained indicate that the applied scheme is highly satisfying and investigate the complicated nonlinear problems that arise in innovation and science. Full article

In this paper, we study some existence and stability results related to the boundary value problem involving the Atangana–Baleanu–Caputo hybrid fractional derivative. More precisely, we transform the studied problem to an equivalent integral equation, and after that, by applying appropriate fixed-point theorems and using suitable conditions, we prove the existence of solutions. Furthermore, we derive sufficient conditions that guarantee the stability in the sense of Hyers–Ulam. To support the theoretical findings, we present illustrative examples along with numerical simulations. Full article

This study presents an analytical method based on the Sumudu transform decomposition method to find an approximate solution for a fractional smoking epidemic model. In this work, fractional derivatives have been taken in the sense of the Yang–Abdel–Cattani operator along with the Atangana–Baleanu derivative in the Caputo sense. A model is developed to explain smoking behavior among adults, which is still a serious health problem worldwide. A nonlinear system is used to study how smoking habit changes with time and how it affects populations. Existence and uniquessness are also derived to show the validity of the approach used. The method applied is simple, stable, and efficient for solving nonlinear fractional epidemic models. Results of the model show that the current model describes the problem in the best possible way and how smoking impacts the population. Full article

This paper examines the sufficient conditions that guarantee the existence of solutions and anti-periodic solutions to five classes of fractional differential equations and inclusions involving the weighted generalized Atangana–Baleanu differential operator of order $\delta \in (1,2)$ under non-local conditions and with instantaneous or non-instantaneous impulses in Banach spaces whose dimension is infinite. First, we deduce some novel properties of this differential operator, then derive the formula for the solutions and anti-periodic solutions, and investigate their existence for the problems presented. Our method relies on certain properties of the Atangana–Baleanu differential operator, which we will obtain, as well as the fixed-point theorems that can be applied to the functions and multi-valued functions. Our work generalizes recently published results. In the final section, we present some examples to illustrate how our theoretical results can be applied. Full article

Nonlinear equations arise extensively in engineering and applied sciences, where efficient and reliable iterative solvers are required. This study introduces two fractional-order iterative schemes based on a common predictor–corrector structure: a Caputo-based method, NCFS₁, and an Atangana–Baleanu–Caputo (ABC)-based variant, ${\mathrm{NFS}}_1^{abc}$. The proposed schemes incorporate a fractional order and two tunable parameters to improve flexibility in the iterative process. The local convergence behavior of the Caputo-based method is analyzed by means of fractional Taylor expansions, yielding an explicit error equation and convergence order, while analogous asymptotic considerations are discussed for the ABC-based variant. A dynamical-systems analysis is also performed through basins of attraction, the Convergence Area Index, and the Wada measure. Numerical experiments on application-motivated nonlinear models indicate that the proposed methods can provide faster error reduction, smaller residuals, and lower computational cost than selected existing fractional iterative schemes. These results suggest that the proposed framework is a flexible and effective approach for nonlinear root-finding problems, combining local convergence analysis with global dynamical assessment. 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

This research presents a fractional-order formulation and mathematical analysis of the Rössler chaotic attractor. By utilizing the Nabla discrete Atangana–Baleanu fractional difference derivative in the Caputo sense, the classical integer-order attractor is extended into the fractional domain. The existence and uniqueness of solutions for the resulting fractional system are established via the fixed-point theorem, thereby ensuring that the recommended attractor is well-posed. Furthermore, the Ulam–Hyers stability is investigated within the Nabla discrete Atangana–Baleanu fractional difference derivative in the Caputo sense framework. For numerical investigations, an Euler numerical scheme adapted to the fractional difference derivative is developed and implemented, yielding high-quality phase portraits of a chaotic attractor. The results highlight the effectiveness of fractional-order modeling and numerical methods in capturing the dynamics and stability of the Rössler chaotic system. Full article

This paper studies a novel nonlinear fractional-order financial stress model involving Atangana–Baleanu–Caputo (ABC) operators. It focuses on memory effects that are both constant and variable. The novelty of the proposed framework lies in combining multiple interconnected channels of systemic stress into one fractional dynamical model and looks at how they change over time and how they respond to sustained external perturbations. Theoretically, we prove well-posedness results and study the equilibrium structure and stability of the given model. On the computational side, we use numerical simulations of the individual stress components and an aggregate systemic stress index to look into short-term dynamics under different memory regimes. We also include a shock-response analysis to show how memory effects change the way stress builds up, relaxes, and spreads when forced. The sensitivity analysis shows that systemic stress is amplified by the forcing and interaction parameters and reduced by the damping parameters. These findings demonstrate that the model provides a new and effective tool for studying systemic financial instability in a fractional setting. 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

This work constructs an orthogonal function system on bounded intervals $[0,R]$ associated with the Atangana–Baleanu–Caputo (ABC) fractional derivative for $\alpha \in (1/2,1)$. Starting from a fractional Laguerre-type equation involving the ABC operator, solutions are obtained via a generalized Frobenius method, yielding series representations with characteristic exponent $\alpha -1$. Rather than postulating a weight function by analogy with classical or Caputo settings, the weight is derived directly from the requirement that the underlying fractional operator be formally self-adjoint on a suitable admissible domain. This operator-theoretic approach leads to the explicit Mittag–Leffler weight $w_{\alpha }\left(x\right)=x^{-(2\alpha -1)}{E}_{\alpha }(-x^{\alpha })$, which intrinsically reflects the nonlocal memory structure of the ABC kernel. A similarity transformation removes the universal singular factor and produces regularized eigenfunctions that are continuous on $[0,R]$ and orthogonal in the weighted $L^2$ space. The weight identity and formal self-adjointness are rigorously verified through a right-Volterra uniqueness argument. Numerical experiments confirm orthogonality up to machine precision, demonstrate spectral convergence for a model ABC differential equation, and illustrate consistency with classical Laguerre polynomials in the limit $\alpha \to 1^{-}$. The resulting framework provides a self-consistent orthogonal system suitable for spectral approximations of problems governed by the ABC operator on bounded domains. Full article

The coupled nonlinear system of fractional Boussinesq–Burger equations that may be utilized to model the propagation of shallow water waves is solved in this study using a novel numerical approach. The fractional derivatives in Caputo–Fabrizio and Atangana–Baleanu manner are executed in the system under consideration. The exact solutions of the proposed nonlinear fractional system are shown in the classical scenario of fractional order at $\mathrm{ß}=1$, whereas the approximate solutions are derived using the natural decomposition method. The series solution is generated such that it is simple to compute. Our results are compared with the exact results which clearly show that the suggested approach solutions quickly converge to the known accurate results. We acquire some analysis of the absolute error by comparing the approximate values with their corresponding precise solutions throughout the provided computations. Numerical and graphical simulations are used to confirm the usefulness of the suggested approach, and the outcomes are compared with well-known methods like the fractional decomposition method (FDM) and Laplace residual power series method (LRPSM). It is evident from the comparison that our approach offers better outcomes compared to other approaches. The results of the suggested method are very accurate and give helpful details on the real dynamics of the proposed system. The obtained outcomes ensure that the suggested approach is more effective and examines the highly nonlinear problems arising in engineering and science. Full article

This paper investigates qualitative properties of fractional delay differential equations formulated in terms of the Atangana–Baleanu–Caputo (ABC) fractional derivative of order $1<\varrho <2$. Three related problem settings are examined: equations with variable delay, the constant-delay case, and a multi-delay extension involving several discrete delay terms. For each formulation, sufficient conditions ensuring existence and uniqueness of solutions are established in both the supremum norm and an exponentially weighted Maksoud norm. The analysis is carried out using Banach’s fixed point theorem in conjunction with progressive contractions and suitable Lipschitz-type conditions. In addition, Ulam–Hyers (UH) and Ulam–Hyers–Rassias (UHR) stability results are derived, providing quantitative estimates on the sensitivity of solutions with respect to perturbations. To complement the theoretical findings, numerical examples are presented, one of which illustrates the behavior of approximate solutions for various fractional orders. Full article

In this study, nonlinear fractional Korteweg–de Vries (KdV) type equations with nonlocal operators are studied using Mittag–Leffler kernels and exponential decay. The KdV equations are well known for its use in modeling ion-acoustic waves in plasma, oceanic dynamics, and shallow-water waves. As a result, mathematicians are working to examine modified and generalized versions of the basic KdV equation. In order to find the solutions of nonlinear fractional KdV equations, an extension of this concept is described in the current paper. The solution of fractional KdV equations is carried out using the well-known natural transform decomposition method (NTDM). To evaluate the problem, we employ the fractional operator in the Caputo–Fabrizio (CF) and the Atangana–Baleanu–Caputo sense (ABC) manner. Nonlinear terms can be handled with Adomian polynomials. The main advantage of this novel approach is that it might offer an approximate solution in the form of convergent series using easy calculations. The dynamical behavior of the resulting solutions have been demonstrated using graphs. Numerical data is represented visually in the tables. The solutions at various fractional orders are found and it is proved that they all tend to an integer-order solution. Additionally, we examine our findings with those of the iterative transform method (ITM) and the residual power series transform method (RPSTM). It is evident from the comparison that our approach offers better outcomes compared to other approaches. The results of the suggested method are very accurate and give helpful details on the real dynamics of each issue. The present technique can be expanded to address other significant fractional order problems due to its straightforward implementation. Full article

The primary objective of this research article is to investigate the concept of extended $\mathcal{F}$-metric spaces and to establish a series of fixed point theorems for generalized contractions within this framework. We further introduce and analyze the notion of interpolative Kannan-type cyclic contractions in extended $\mathcal{F}$-metric spaces, deriving several novel fixed point results associated with these mappings. In addition, we obtain common fixed point theorems for rational contractions, thereby extending and unifying a variety of existing results available in the literature. To highlight the novelty and effectiveness of the proposed results, several illustrative examples are provided. Moreover, the theoretical findings are successfully applied to the solution of Atangana–Baleanu fractional integral equations as well as Volterra integral equation of Hammerstein type, demonstrating their practical significance and wide-ranging applicability. Full article