Search Results (150)

The interplay between special functions and geometric function theory continues to inspire significant advances in the study of analytic and multivalent functions. In this work, we introduce and investigate several new subfamilies of multivalent functions associated with the generalized Mittag-Leffler-type Poisson distribution in the open unit disk. We establish necessary and sufficient conditions characterizing membership in these classes and derive meaningful inclusion relationships among them. Furthermore, we define a novel integral operator linked to the generalized Mittag-Leffler-type Poisson distribution and examine its mapping properties and structural connections with the proposed function classes. The results presented herein not only unify and extend a variety of earlier contributions but also demonstrate the effectiveness of distribution-theoretic methods in the analysis of multivalent functions. Full article

A non-homogeneous fractional differential equation with a variable coefficient involving a Caputo fractional derivative is considered. The equation is first transformed into an integral equation and then solved using the method of successive approximations. The obtained general solution involves a generalized Mittag–Leffler-type function and Meijer G-functions. Example solutions corresponding to particular choices of the non-homogeneous term are presented. As an application of the considered non-homogeneous equation, direct and inverse source problems are studied. The solutions are expressed in the form of series expansions using an orthogonal basis obtained through separation of variables. Illustrative examples for the direct and inverse problems are also presented for specific choices of the initial and final time data and the source function. Full article

This paper investigates the synchronization problem for a class of proportional Caputo fractional-order neural networks with respect to another function. A master–slave framework is formulated, and a linear state-feedback controller is proposed for the response system. Under a standard Lipschitz condition on the activation functions, sufficient conditions ensuring the convergence of the synchronization error to zero are established. The analysis is based on an explicit integral representation of the error system, a generalized Gronwall-type inequality, and asymptotic properties of the Mittag–Leffler function. The obtained criterion explicitly reveals the roles of the fractional order, the proportional parameter, the control gain, and the network interconnection matrix. Numerical experiments based on a benchmark fractional Hopfield neural network illustrate the effectiveness of the proposed approach. In particular, a scaled benchmark satisfying all theoretical assumptions provides a strict validation of the main theorem, while the original benchmark highlights the conservative nature of the derived sufficient conditions. Full article

A new generalization of the Logarithmic mean function and Euler’s Beta k-Logarithm function is proposed using the Mittag–Leffler k-function. We study their analytical properties, including functional relations, symmetry relation, inequalities, summation representations, and integral representations. Mellin transformations are established, and a generalized k-Beta Logarithmic distribution is presented along with its probabilistic applications. Furthermore, we introduce a novel k-Beta Logarithmic fractional derivative operator of Caputo type and develop a Legendre spectral collocation method with Chebyshev–Gauss–Lobatto nodes for the numerical solution of associated fractional differential equations. Rigorous error analysis in the weighted $L^2$-norm is provided, establishing algebraic convergence for finite-regularity solutions and exponential convergence for analytic solutions. Numerical experiments confirm the theoretical convergence rates and demonstrate the efficiency and spectral accuracy of the proposed scheme. 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

The Mittag-Leffler function holds significant importance in fractional calculus due to its extensive applications in addressing challenges across science, engineering, biology, hydrology, and earth sciences. Notably, the closed-form solution of a time-fractional model naturally emerges as the Mittag-Leffler function (MLF), necessitating precise and efficient computations. Consequently, numerical approximations are essential for accurately calculating the Mittag-Leffler function. In this study, we develop a straightforward yet precise real pole rational approximation for the Mittag-Leffler function. We demonstrate first-order convergence and L-acceptability, which aid in mitigating unwanted oscillations. Additionally, we create an effective and precise first-order generalized exponential time differencing scheme to solve the time-fractional reaction–diffusion equations. We obtain and prove the convergence result using Grönwall-type inequality. Several numerical experiments are conducted to confirm the efficiency and accuracy of the proposed numerical scheme compared with exact solutions. The computational efficiency of the proposed method is compared with another existing first-order numerical technique. Furthermore, our proposed scheme is crucial for developing higher-order predictor–corrector schemes for solving time-fractional models. Full article

Generalized incommensurate fractional differential systems (GIFDSs) unify classical fractional frameworks via weight functions, capturing non-uniform multicomponent system dynamics. This paper fills a critical research gap by analyzing GIFDSs for both commensurate and incommensurate weight functions. For commensurate weights ($w_i\left(t\right)=w\left(t\right)$), classical IFDS equivalence is established via state transformation. Linear homogeneous mild solutions are derived using the incommensurate Mittag–Leffler function. Existence and uniqueness of nonlinear solutions are proved under continuity and Lipschitz assumptions. Hyers–Ulam stability is verified for linear non-homogeneous systems. For incommensurate weights (distinct $w_i\left(t\right))$, a novel framework is developed: by the integral bound lemma and Picard iteration, local existence (existence on $[a,t_1]$) is established, then it is extended to the full interval. The global uniqueness is obtained by Gronwall-type inequality via combined substitution. These results are applied to Hopfield Neural Networks, showing that one-layer HNNs with tanh or sigmoid activations admit unique mild solutions under GIFDS dynamics. Full article

A continuous-time Markov chain framework is developed for service life prediction of building assets, and three formulations are compared: a homogeneous generator, a time-varying generator, and a fractional model. The framework delivers survival, density of absorption time, hazard, and mean time to absorption. For the homogeneous case, state trajectories are computed using matrix exponentials. The time-varying case is solved both by local exponential propagation on a time grid and by direct integration of the Kolmogorov equation. The fractional case is implemented in two independent ways, via a truncated series expansion and via an in-house routine for the Mittag-Leffler function, which also allows the direct evaluation of survival and hazard from the standard fractional relations while avoiding singular behaviour at the origin. This study shows that non-homogeneous rates accelerate deterioration relative to the homogeneous benchmark, whereas fractional dynamics reproduce early-time acceleration followed by a slow decline of the hazard, which is consistent with heavy-tailed survival and longer effective service life. The two fractional solvers provide mutually consistent outputs, which supports the numerical robustness of the approach. The framework is readily applicable to sparse inspection data and short observation windows and provides a transparent basis for comparing modelling assumptions that affect life cycle forecasts used in asset management and maintenance planning. Full article

In this work, we propose a dynamic cobweb-type market equilibrium model where supply responds to price using a Mittag–Leffler kernel generalized weighted Caputo fractional derivative. By permitting independent fractional orders, a time-varying weight function that reweights historical data, and a monotone economic-time transformation that captures heterogeneous adjustment speeds, the formulation expands upon previous fractional cobweb models. We begin by highlighting several special cases encompassed by our proposed model. Next, we establish well-posedness, covering existence, uniqueness, and continuous dependence on initial data and parameters via an equivalent Volterra integral formulation, alongside a positivity theorem that ensures prices remain economically meaningful. Then, we derive stability conditions for the perturbation dynamics and characterize the constant equilibrium price. To perform the simulation, we constructed an explicit Volterra partition scheme specifically designed for the generalized kernel and established its convergence. In addition, we validated this approach using numerical examples illustrating how fractional orders, weights, and time transformations cause transient oscillations and convergence toward equilibrium. Full article

Fractional calculus approach is used to analyze the model of surfactant transport by anomalous diffusion and its adsorption on an interface in a liquid-liquid system. The anomalous diffusion is modeled by time-fractional partial differential equations in the bulk phases. The adsorption of surfactant is described by the corresponding time-fractional Neumann boundary conditions at the interface. The adsorption process is considered under mixed barrier-diffusion control, described by first-order ordinary differential equation, which relates the subsurface concentration with that on the interface. A second relation between these concentrations is derived in terms of a fractional equation by application of Laplace transform technique. By combining both relations the subsurface concentration is eliminated and a single multi-term fractional ordinary differential equation for the surfactant concentration on the interface is derived. Different adsorption kinetic models are considered. In the case of Henry adsorption isotherm the model is linear and possesses analytical solution in terms of multinomial Mittag-Leffler functions. In the cases of Volmer and van der Waals adsorption isotherms nonlinear differential equations of fractional order are obtained. They are reformulated in equivalent integral form, which is used for computer simulation of the process of adsorption. Numerical results are presented and compared with analytical asymptotic predictions. 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

The current manuscript is concerned with the uniqueness and existence of a solution for a new class of $\Phi$-Hilfer fractional neutral functional integro-differential equations ($\Phi$-HFNFIDEs) with terminal conditions. Firstly, employing Babenko’s approach, we convert the aforesaid equation under consideration into an analogous integral equation. More precisely, using the multivariate Mittag-Leffler function, Banach contraction principle, and Krasnoselskii’s fixed-point theorem, we derive some conditions that guarantee the uniqueness and the existence of the solutions. For an illustration of our results in this manuscript, two examples are provided as well. Full article

This work introduces a new category of proportional fractional integro-differential equations (PFIDEs) governed by functional boundary conditions. We derive verifiable sufficient criteria that guarantee the Ulam–Hyers Stability, existence and uniqueness of solutions to this problem. Our analytical approach leverages Babenko’s method to construct an inverse operator, which allows us to reformulate the differential problem into an equivalent integral equation. The analysis is then conducted using key mathematical tools, including contraction mapping principle of Banach, the Leray–Schauder alternative, and properties of multivariate Mittag–Leffler functions. The Ulam–Hyers Stability is rigorously examined to assess the system’s resilience to small perturbations. The applicability and effectiveness of the established theoretical results are demonstrated through two illustrative examples. This research provides a unified and adaptable framework that advances the analysis of complex fractional-order dynamical systems subject to nonlocal constraints. Full article

Quantum calculus is a powerful extension of classical calculus, providing novel tools for deriving sharper and more efficient analytical results without relying on limits. This study investigates error estimations for Milne formula-type inequalities within the framework of quantum calculus, offering a fresh perspective on numerical integration theory. New variants of Milne’s formula-type inequalities are established for q-differentiable convex functions by first deriving a key quantum integral identity. The primary aim of this work is to obtain sharper and more accurate bounds for Milne’s formula compared to existing results in the literature. The validity of the proposed results is demonstrated through illustrative examples and graphical analysis. Furthermore, applications to special means of real numbers, the Mittag–Leffler function, and numerical integration formulas are presented to emphasize the practical significance of the findings. This study contributes to advancing the theoretical foundations of both classical and quantum calculus and enhances the understanding of integral inequality theory. Full article

In this paper, we evaluate a general class of finite integrals involving the error function, generalized Mittag-Leffler functions, and incomplete Aleph functions. The main result provides a unified framework that extends several known formulas related to the incomplete Gamma, I-, and H-functions. Under suitable conditions, these results reduce to many classical special cases. We discuss convergence conditions that justify the validity of the obtained formulas and include explicit corollaries that highlight connections with earlier results in the literature. To illustrate applicability, we present numerical examples and graphs, demonstrating the behavior of the error function integral and Mittag-Leffler functions for specific parameter values. These integrals arise naturally in fractional calculus, probability theory, viscoelasticity, and anomalous diffusion, underscoring the importance of the present work in both mathematical analysis and applications. Full article