Search Results (438)

The Riemann-Liuoville fractional integrals are the simplest and most popular operators of the classical fractional calculus. But their variants, the Erdélyi-Kober operators of fractional integration, have many more applications due to the freedom to choose the additional (three) parameters. We introduce and study a generalization of the Erdélyi-Kober and Riemann-Liuoville fractional integrals, where the elementary kernel function is replaced by a suitably chosen $I_{1,1}^{1,0}$-function. The I-functions introduced by Rathie in 1997 are generalized hypergeometric functions extending the Fox H-functions and the Meijer G-functions. Note that till recently this new class of special functions has not been popular because of their too complicated structure involving fractional powers of the Gamma functions and their multi-valued behavior. However, the I-functions happened to arise not only for the needs of statistical physics, but also since they included important special functions in mathematics that were not covered by the H- and G-functions. In our previous works, as Kiryakova and Paneva-Konovska, we have shown the relations of such functions, among which are the Mittag-Leffler and Le Roy type, their multi-index variants, and others related to fractional calculus, to the I-functions. Here, we propose a new theory of generalization of the Erdélyi-Kober fractional integrals, based on the use of an I-function as a kernel. This will serve next as a base to extend our generalized multi-order fractional calculus with operators involving $I_{m,m}^{m,0}$. In this paper, we also evaluate the images under these new generalized fractional integrals of special functions of very general form. Finally, in the Conclusion section, we comment on some earlier discussions on the relations between fractal geometry and fractional calculus, nowadays already without any doubts. Full article

In this study, we consider a fractional-order extension of the Jeffreys equation (also known as the dual-phase-lag equation) by introducing the Reimann–Liouville fractional integral, of order $0<\nu <1$, to the Jeffreys constitutive law, where for $\nu =1$ it corresponds to the conventional Jeffreys equation. The kinetical behaviors of the fractional equation such as non-negativity of the propagator, mean-squared displacement, and the temporal amplitude are investigated. The fractional Langevin equation, or the fractional damped oscillator, is a special case of the considered integrodifferential equation governing the temporal amplitude. When $\nu =0$ and $\nu =1$, the fractional differential equation governing the temporal amplitude has the mathematical structure of the classical linear damped oscillator with different coefficients. The existence of a real solution for the new temporal amplitude is proven by deriving this solution using the complex integration method. Two forms of conditional closed-form solutions for the temporal amplitude are derived in terms of the Mittag–Leffler function. It is found that the proposed generalized fractional damped oscillator equation results in underdamped oscillations in the case of $0<\nu <1$, under certain constraints derived from the non-fractional case. Although the nonfractional case has the form of classical linear damped oscillator, it is not necessary for its solution to have the three common types of oscillations (overdamped, underdamped, and critical damped), unless a certain condition is met on the coefficients. The obtained results could be helpful for analyzing thermal wave behavior in fractals, heterogeneous materials, or porous media since the fractional-order derivatives are related to the porosity of media. Full article

In this article, a novel type of equation, namely the $\mathsf{\Phi }$-Hilfer fractional-order integro-differential delay system ($\mathsf{\Phi }$-HFOIDDS), is proposed. Here, we study the existence and Hyers–Ulam–Mittag–Leffler (H-U-M-L) stability of the aforementioned equation which are obtained by using the multivariate Mittag–Leffler function, Banach contraction principle, and Picard operator method as well as generalized Gronwall inequality. Finally, we conclude this paper by constructing a suitable example to illustrate the applicability of the principal outcomes. Full article

In the present scenario, coherent states of a quantum harmonic oscillator are generalized with positive Fox H auxiliary functions. The novel generalized coherent states provide canonical coherent states and Mittag-Leffler or Wright generalized coherent states, as particular cases, and resolve the identity operator, over the Fock space, with a weight function that is the product of a Fox H function and a Wright generalized hypergeometric function. The novel generalized coherent states, or the corresponding truncated generalized coherent states, are characterized by anomalous statistics for large values of the number of excitations: the corresponding decay laws exhibit, for determined values of the involved parameters, various behaviors that depart from exponential and inverse-power-law decays, or their product. The analysis of the Mandel Q factor shows that, for small values of the label, the statistics of the number of excitations becomes super-Poissonian, or sub-Poissonian, by simply choosing sufficiently large values of one of the involved parameters. The time evolution of a generalized coherent state interacting with a thermal reservoir and the purity are analyzed. Full article

In this paper, we primarily use the inverse operator method to find a unique series solution to a time–fractional wave equation with variable coefficients based on the Mittag–Leffler function. In addition, we also derive the series and integral convolution solutions to the Klein–Gordon equation using the Fourier transform and Green’s functions. Furthermore, our series solutions significantly simplify the process of finding solutions with several illustrative examples, avoiding the need for complicated integral computations. Full article

The asymptotic behavior of discrete Riemann–Liouville fractional difference equations is a fundamental problem with important mathematical and physical implications. In this paper, we investigate a particular case of such an equation of the order $0.5$ subject to a given initial condition. We establish the existence of a unique solution expressed via a Mittag-Leffler-type function. The delta-asymptotic behavior of the solution is examined, and its convergence properties are rigorously analyzed. Numerical experiments are conducted to illustrate the qualitative features of the solution. Furthermore, a neural network-based approximation is employed to validate and compare with the analytical results, confirming the accuracy, stability, and sensitivity of the proposed method. Full article

This paper investigates the Ulam–Hyers stability of both linear and nonlinear delayed neutral Hilfer fractional difference equations. We utilize the nabla Laplace transform, known as the N-transform, along with a generalized discrete Gronwall inequality to derive sufficient conditions for stability. For the linear case, we provide an explicit solution formula involving discrete Mittag-Leffler functions and establish its stability properties. In the nonlinear case, we concentrate on delayed neutral Hilfer fractional difference equations, a class of systems that appears to be unexplored in the existing literature with respect to Ulam–Hyers stability. In particular, for the linear case, the absolute difference between the solution of the linear Hilfer fractional difference equation and the solution of the corresponding perturbed equation is bounded by the function of $\epsilon$ when the perturbed term is bounded by $\epsilon$. In the case of the neutral fractional delayed Hilfer difference equation, the absolute difference is bounded by a constant multiple of $\epsilon$. Our results fill this gap by offering novel stability criteria. We support our theoretical findings with illustrative numerical examples and simulations, which visually confirm the predicted stability behavior and demonstrate the applicability of the results in discrete fractional dynamic systems. Full article

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

A mathematical model of the anomalous diffusion of surfactant and the process of adsorption–desorption on an interface is analyzed using a fractional calculus approach. The model is based on time-fractional partial differential equations in the bulk phases and the corresponding time-fractional description of the flux bulk–interface. The general case, when the surfactant is soluble in both phases, is considered under the assumption that the adsorption–desorption process is diffusion-controlled. Some of the most popular kinetic models of Henry, Langmuir, and Volmer are considered. Applying the Laplace transform, the partial differential model is transformed into a single multi-term time-fractional nonlinear ordinary differential equation for the surfactant concentration on the interface. Based on existing analytical solutions of linear time-fractional differential equations, the exact solution in the case of the Henry model is derived in terms of multinomial Mittag–Leffler functions, and its asymptotic behavior is studied. Further, the fractional differential model in the general nonlinear case is rewritten as an integral equation, which is a generalization of the well-known Ward–Tordai equation. For computer simulations, based on the obtained integral equation, a predictor–corrector numerical technique is developed. Numerical results are presented and analyzed. Full article

In this paper, we investigate functional inverse operators associated with a class of fractional integro-differential equations. We further explore the existence, uniqueness, and stability of solutions to a new integro-differential equation featuring variable coefficients and a functional boundary condition. To demonstrate the applicability of our main theorems, we provide several examples in which we compute values of the two-parameter Mittag–Leffler functions. The proposed approach is particularly effective for addressing a wide range of integral and fractional nonlinear differential equations with initial or boundary conditions—especially those involving variable coefficients, which are typically challenging to treat using classical integral transform methods. Finally, we demonstrate a significant application of the inverse operator approach by solving a Caputo fractional convection partial differential equation in ${\mathbb{R}}^n$ with an initial condition. Full article

This article is devoted to the derivation of the Laplace transforms of the derivatives with respect to parameters of certain special functions, namely, the Mittag–Leffler-type, Wright, and Le Roy-type functions. These formulas show the interconnection of these functions and lead to a better understanding of their behavior on the real line. These formulas are represented in a convoluted form and reconstructed in a more suitable form by using the Efros theorem. Full article

This paper investigates the synchronization control problem for delayed fractional-order neural networks (DFONNs) with mismatched parameters. A novel synchronization behavior termed quasi-Mittag-Leffler projective synchronization (QMLPS) is studied. The core contribution of this work lies in the following: (1) The time delay and mismatched parameters between driven and response systems are considered, which is more general. Both static controllers and adaptive controllers are designed to synchronize the DFONNs. (2) The synchronization errors are estimated, and the rate of convergence is clarified description. By using the Lyapunov stability theory and some significant fractional-order differential inequalities, some sufficient conditions for DFONNs are derived under two kinds of control methods; furthermore, the bound of synchronization errors is estimated by the Mittag-Leffler function. Quantitative numerical simulations have demonstrated the superiority of our controller. Compared to existing results, the QMLPS introduced in this paper is more general, incorporating many existing synchronization concepts. The numerical simulation section verifies the effectiveness of the theoretical results, providing several types of synchronization behaviors of the controlled system under both mismatched and matched parameter conditions, and it also demonstrates the accuracy of the theoretical estimation of synchronization error bounds. Full article

The study of various subclasses of univalent functions involving the solutions to various differential equations is not totally new, but studies of analytic functions with respect to $(\mathsf{\ell },\kappa )$-symmetric points are rarely conducted. Here, using a differential operator which was defined using the Hadamard product of Mittag–Leffler function and general analytic function, we introduce a new class of starlike functions with respect to $(\mathsf{\ell },\kappa )$-symmetric points associated with the vertical domain. To define the function class, we use a Carathéodory function which was recently introduced to study the impact of various conic regions on the vertical domain. We obtain several results concerned with integral representations and coefficient inequalities for functions belonging to this class. The results obtained by us here not only unify the recent studies associated with the vertical domain but also provide essential improvements of the corresponding results. Full article

This study introduces an innovative analytical solution to the time-fractional Cattaneo heat conduction equation, which models photothermal transport in metallic thin films subjected to short laser pulse irradiation. The model integrates the Caputo fractional derivative of order 0 < p ≤ 1, addressing non-Fourier heat conduction characterized by finite wave speed and memory effects. The equation is nondimensionalized through suitable scaling, incorporating essential elements such as a newly specified laser absorption coefficient and uniform initial and boundary conditions. A hybrid approach utilizing the finite Fourier cosine transform (FFCT) in spatial dimensions and the Laplace transform in temporal dimensions produces a closed-form solution, which is analytically inverted using the two-parameter Mittag–Leffler function. This function inherently emerges from fractional-order systems and generalizes traditional exponential relaxation, providing enhanced understanding of anomalous thermal dynamics. The resultant temperature distribution reflects the spatiotemporal progression of heat from a spatially Gaussian and temporally pulsed laser source. Parametric research indicates that elevating the fractional order and relaxation time amplifies temporal damping and diminishes thermal wave velocity. Dynamic profiles demonstrate the responsiveness of heat transfer to thermal and optical variables. The innovation resides in the meticulous analytical formulation utilizing a realistic laser source, the clear significance of the absorption parameter that enhances the temperature amplitude, the incorporation of the Mittag–Leffler function, and a comprehensive investigation of fractional photothermal effects in metallic nano-systems. This method offers a comprehensive framework for examining intricate thermal dynamics that exceed experimental capabilities, pertinent to ultrafast laser processing and nanoscale heat transfer. 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