Search Results (47)

In this paper, we derive novel results concerning third-order differential subordinations for meromorphic functions, utilizing a newly defined linear operator that involves the inverse of the Legendre chi function in conjunction with the Mittag-Leffler identity. To establish these results, we introduce several families of admissible functions tailored to this operator and formulate sufficient conditions under which the subordinations hold. Our study presents three fundamental theorems that extend and generalize known results in the literature. Each theorem is accompanied by rigorous proofs and further supported by corollaries and illustrative examples that validate the applicability and sharpness of the derived results. In particular, we highlight special cases and discuss their implications through both analytical evaluations and graphical interpretations, demonstrating the strength and flexibility of our framework. This work contributes meaningfully to the field of geometric function theory by offering new insights into the behavior of third-order differential operators acting on p-valent meromorphic functions. Furthermore, the involvement of the Mittag-Leffler function positions the results within the broader context of fractional calculus, suggesting potential for applications in the mathematical modeling of complex and nonlinear phenomena. We hope this study stimulates further research in related domains. Full article

This paper introduces a novel version of the Gronwall inequality specifically related to the Hilfer–Hadamard proportional fractional derivative. By utilizing Picard’s method of successive approximations along with the definition of Mittag–Leffler functions, we derive a representation formula for the solution of the Hilfer–Hadamard proportional fractional differential equation featuring constant coefficients, expressed in the form of the Mittag–Leffler kernel. We establish the uniqueness of the solution through the application of Banach’s fixed-point theorem, leveraging several properties of the Mittag–Leffler kernel. The current study outlines optimal stability, a new Ulam-type concept based on classical special functions. It aims to improve approximation accuracy by optimizing perturbation stability, offering flexible solutions to various fractional systems. While existing Ulam stability concepts have gained interest, extending and optimizing them for control and stability analysis in science and engineering remains a new challenge. The proposed approach not only encompasses previous ideas but also emphasizes the enhancement and optimization of model stability. The numerical results, presented in tables and charts, are provided in the application section to facilitate a better understanding. Full article

In this paper, we aim to extend the bicomplex two-parameter Mittag-Leffler (M-L) function by introducing a new k-parameter. This results in the definition of the bicomplex k-M-L function with two parameters. This generalization offers more flexibility and broader applicability in modeling complex fractional systems. We explore its key properties, develop new theorems, and establish the corresponding k-Riemann–Liouville fractional calculus within the bicomplex setting for the extended function. Furthermore, we solve several fractional differential equations using the bicomplex k-M-L function with two parameters. The results prove the enhanced flexibility and generality of the proposed function, particularly in deriving fractional kinetic equations, offering novel insights beyond existing bicomplex fractional models. Full article

In this paper, a generalized acoustic black hole (ABH) beam covered with a viscoelastic layer is proposed to improve the energy dissipation based on the double-parameter Mittag–Leffler (ML) function. Since fractional-order constitutive models can more accurately capture the properties of viscoelastic materials, a fractional dynamic model of an ABH structure covered with viscoelastic film is established based on the fractional Kelvin–Voigt constitutive equation and the mechanical analysis of composite structures. To analyze the energy dissipation of the viscoelastic ML-ABH structures under steady-state conditions, the wave method is introduced, and the theory of vibration wave transmission in such non-uniform structures is extended. The effects of the fractional order, the film thickness and length, and shape function parameters on the dynamic characteristics of the ABH structure are systematically investigated. The study reveals that these parameters have a significant impact on the vibration characteristics of the ABH structure. To obtain the best parameters of the shape function under various parameters, the Particle Swarm Optimization (PSO) algorithm is employed. The results demonstrate that by selecting appropriate ML parameters and viscoelastic materials, the dissipation characteristics of the structure can be significantly improved. This research provides a theoretical foundation for structural vibration reduction in ABH structures. Full article

We introduce a novel class of coherent states, termed ${\mathcal{W}}_{\alpha ,\beta ,\nu }$-coherent states, constructed using a deformed boson algebra based on the generalised factorial ${\left[n\right]}_{\alpha ,\beta ,\nu }!$. This algebra extends conventional factorials, incorporating advanced special functions such as the Mittag-Leffler and Wright functions, enabling the exploration of a broader class of quantum states. The mathematical properties of these states, including their continuity, completeness, and quantum fluctuations, are analysed. A key aspect of this work is the resolution of the Stieltjes moment problem associated with these states, achieved through the inverse Mellin transformation method. The framework provides insights into the interplay between the classical and quantum regimes, with potential applications in quantum optics and fractional quantum mechanics. By extending the theoretical landscape of coherent states, this study opens avenues for further exploration in mathematical physics and quantum technologies. Full article

The aim of this research article is to use the extended fractional operators involving the multivariate Mittag–Leffler (M-M-L) function, we provide the generalization of the Hermite–Hadamard–Fejer (H-H-F) inequalities. We relate these inequalities to previously published disparities in the literature by making appropriate substitutions. In the last section, we analyze several inequalities related to the H-H-F inequalities, focusing on generalized h-convexity associated with extended fractional operators involving the M-M-L function. To achieve this, we derive two identities for locally differentiable functions, which allows us to provide specific estimates for the differences between the left, middle, and right terms in the H-H-F inequalities. Also, we have constructed specific inequalities and visualized them through graphical representations to facilitate their applications in analysis. The research bridges theoretical advancements with practical applications, providing high-accuracy bounds for complex systems involving fractional calculus. Full article

We show how a rescaling of fractional operators with bounded kernels may help circumvent their documented deficiencies, for example, the inconsistency at zero or the lack of inverse integral operator. On the other hand, we build a novel class of linear operators with memory effects to extend the L-fractional and the ordinary derivatives, using probability tools. A Mittag–Leffler-type function is introduced to solve linear problems, and nonlinear equations are addressed with power series, illustrating the methods for the SIR epidemic model. The inverse operator is constructed, and a fundamental theorem of calculus and an existence-and-uniqueness result for differintegral equations are proven. A conjecture on deconvolution is raised, which would permit completing the proposed theory. Full article

In this paper, we consider and study in detail the generalized Fox–Wright function ${}_p\tilde{\Psi }_q$ introduced in our recent work as an extension of the Fox–Wright function ${}_p\Psi _q$. This special function can be seen as an important case of the so-called I-functions of Rathie and $\overline{H}$-functions of Inayat-Hussain, that in turn extend the Fox H-functions and appear to include some Feynman integrals in statistical physics, in polylogarithms, in Riemann Zeta-type functions and in other important mathematical functions. Depending on the parameters, ${}_p\tilde{\Psi }_q$ is an entire function or is analytic in an open disc with a final radius. We derive its basic properties, such as its order and type, and its images under the Laplace transform and under classical fractional-order integrals. Particular cases of ${}_p\tilde{\Psi }_q$ are specified, including the Mittag-Leffler and Le Roy-type functions and their multi-index analogues and many other special functions of Fractional Calculus. The corresponding results are illustrated. Finally, we emphasize the role of these new generalized hypergeometric functions as eigenfunctions of operators of new Fractional Calculus with specific I-functions as singular kernels. This paper can be considered as a natural supplement to our previous surveys “Going Next after ‘A Guide to Special Functions in Fractional Calculus’: A Discussion Survey”, and “A Guide to Special Functions of Fractional Calculus”, published recently in this journal. Full article

The two-parameter Mittag–Leffler function $E_{\alpha ,\beta }$ is of fundamental importance in fractional calculus, and it appears frequently in the solutions of fractional differential and integral equations. However, the expense of calculating this function often prompts efforts to devise accurate approximations that are more cost-effective. When $\alpha >1$, the monotonicity property is largely lost, resulting in the emergence of roots and oscillations. As a result, current rational approximants constructed mainly for $\alpha \in (0,1)$ often fail to capture this oscillatory behavior. In this paper, we develop computationally efficient rational approximants for $E_{\alpha ,\beta }(-t)$, $t\ge 0$, with $\alpha \in (1,2)$. This process involves decomposing the Mittag–Leffler function with real roots into a weighted root-free Mittag–Leffler function and a polynomial. This provides approximants valid over extended intervals. These approximants are then extended to the matrix Mittag–Leffler function, and different implementation strategies are discussed, including using partial fraction decomposition. Numerical experiments are conducted to illustrate the performance of the proposed approximants. Full article

The planning, regulation and effectiveness of the product design process depend on various characteristics. Recently, bio-inspired collective intelligence approaches have been applied in this process in order to create more appealing product forms and optimize the design process. In fact, the use of neural network models in product form design analysis is a complex process, in which the type of network has to be determined, as well as the structure of the network layers and the neurons in them; the connection coefficients, inputs and outputs have to be explored; and the data have to be collected. In this paper, an impulsive discrete fractional neural network modeling approach is introduced for product design analysis. The proposed model extends and complements several existing integer-order neural network models to the generalized impulsive discrete fractional-order setting, which is a more flexible mechanism to study product form design. Since control and stability methods are fundamental in the construction and practical significance of a neural network model, appropriate impulsive controllers are designed, and practical Mittag-Leffler stability criteria are proposed. The Lyapunov function strategy is applied in providing the stability criteria and their efficiency is demonstrated via examples and a discussion. The established examples also illustrate the role of impulsive controllers in stabilizing the behavior of the neuronal states. The proposed modeling approach and the stability results are applicable to numerous industrial design tasks in which multi-agent systems are implemented. Full article

Ultrafast diffusion disperses faster than super-diffusion, and this has been proven by several theoretical and experimental investigations. The mean square displacement of ultrafast diffusion grows exponentially, which provides a significant challenge for modeling. Due to the inhomogeneity, nonlinear interactions, and high porosity of cement materials, the motion of particles on their surfaces satisfies the conditions for ultrafast diffusion. The investigation of the diffusion behavior in cementitious materials is crucial for predicting the mechanical properties of cement. In this study, we first attempted to investigate the dynamic of ultrafast diffusion in cementitious materials underlying the Riemann–Liouville nonlocal structural derivative. We constructed a Riemann–Liouville nonlocal structural derivative ultrafast diffusion model with an exponential function and then extended the modeling strategy using the Mittag–Leffler function. The mean square displacement is analogous to the integral of the corresponding structural derivative, providing a reference standard for the selection of structural functions in practical applications. Based on experimental data on cement mortar, the accuracy of the Riemann–Liouville nonlocal structural derivative ultrafast diffusion model was verified. Compared to the power law diffusion and the exponential law diffusion, the mean square displacement with respect to the Mittag–Leffler law is closely tied to the actual data. The modeling approach based on the Riemann–Liouville nonlocal structural derivative provides an efficient tool for depicting ultrafast diffusion in porous media. Full article

In this paper, we obtain the existence and uniqueness theorem for solutions of Caputo-type fractional stochastic delay differential systems(FSDDSs) with Poisson jumps by utilizing the delayed perturbation of the Mittag–Leffler function. Moreover, by using the Burkholder–Davis–Gundy inequality, Doob’s martingale inequality, and Hölder inequality, we prove that the solution of the averaged FSDDSs converges to that of the standard FSDDSs in the sense of $L^p$. Some known results in the literature are extended. Full article

In the survey Kiryakova: “A Guide to Special Functions in Fractional Calculus” (published in this same journal in 2021) we proposed an overview of this huge class of special functions, including the Fox H-functions, the Fox–Wright generalized hypergeometric functions ${}_p{\Psi }_q$ and a large number of their representatives. Among these, the Mittag-Leffler-type functions are the most popular and frequently used in fractional calculus. Naturally, these also include all “Classical Special Functions” of the class of the Meijer’s G- and ${}_p{F}_q$-functions, orthogonal polynomials and many elementary functions. However, it so happened that almost simultaneously with the appearance of the Mittag-Leffler function, another “fractionalized” variant of the exponential function was introduced by Le Roy, and in recent years, several authors have extended this special function and mentioned its applications. Then, we introduced a general class of so-called (multi-index) Le Roy-type functions, and observed that they fall in an “Extended Class of SF of FC”. This includes the I-functions of Rathie and, in particular, the $\overline{H}$-functions of Inayat-Hussain, studied also by Buschman and Srivastava and by other authors. These functions initially arose in the theory of the Feynman integrals in statistical physics, but also include some important special functions that are well known in math, like the polylogarithms, Riemann Zeta functions, some famous polynomials and number sequences, etc. The I- and $\overline{H}$-functions are introduced by Mellin–Barnes-type integral representations involving multi-valued fractional order powers of $\Gamma$-functions with a lot of singularities that are branch points. Here, we present briefly some preliminaries on the theory of these functions, and then our ideas and results as to how the considered Le Roy-type functions can be presented in their terms. Next, we also introduce Gelfond–Leontiev generalized operators of differentiation and integration for which the Le Roy-type functions are eigenfunctions. As shown, these “generalized integrations” can be extended as kinds of generalized operators of fractional integration, and are also compositions of “Le Roy type” Erdélyi–Kober integrals. A close analogy appears with the Generalized Fractional Calculus with H- and G-kernel functions, thus leading the way to its further development. Since the theory of the I- and $\overline{H}$-functions still needs clarification of some details, we consider this work as a “Discussion Survey” and also provide a list of open problems. Full article

In this paper, we present a novel class of analytic functions in the form $h\left(z\right)=z^p+{\displaystyle \sum _{k=p+1}^{\infty }}{a}_k{z}^k$ in the unit disk. These functions establish a connection between the extended Mittag–Leffler function and the quantum operator presented in this paper, which is denoted by ${\aleph }_{q,p}^n{(}_{\mathcal{L}},a,b)$ and is also an extension of the Raina function that combines with the Jackson derivative. Through the application of differential subordination methods, essential properties like bounds of coefficients and the Fekete–Szegő problem for this class are derived. Additionally, some results of special cases to this study that were previously studied were also highlighted. Full article

We present a systematic introduction to a class of functions that provide fundamental solutions for autonomous linear integer-order and fractional-order delay differential equations. These functions, referred to as delay functions, are defined through power series or fractional power series, with delays incorporated into their series representations. Using this approach, we have defined delay exponential functions, delay trigonometric functions and delay fractional Mittag-Leffler functions, among others. We obtained Laplace transforms of the delay functions and demonstrated how they can be employed in finding solutions to delay differential equations. Our results, which extend and unify previous work, offer a consistent framework for defining and using delay functions. Full article