Search Results (34)

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 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

This paper investigates the fractional calculus of distributions supported on ${\mathbb{R}}^{+}$ in the sense of L. Schwartz, based on distributional convolutions. We further study a generalized Bagley–Torvik equation involving an arbitrary number of fractional derivative terms with orders in the interval $(0,2)$. The existence and uniqueness of solutions for its nonlinear form are established in a space of continuous functions by applying Banach’s contraction principle, the Leray–Schauder fixed-point theorem, inverse operators, and the multivariate Mittag–Leffler function. Finally, several examples are presented, in which the values of multivariate Mittag–Leffler functions are computed to illustrate the main results. Full article

This paper presents a unified analytical and computational framework for the study of typhoid fever transmission dynamics governed by a Caputo fractional-order compartmental model of order $\kappa \in (0,1]$. The population is stratified into five epidemiological classes, namely susceptible $\left(\mathcal{S}\right)$, asymptomatic $\left(\mathcal{A}\right)$, symptomatic $\left(\mathcal{I}\right)$, hospitalised $\left(\mathcal{H}\right)$, and recovered $\left(\mathcal{R}\right)$, and the governing system explicitly incorporates asymptomatic transmission, treatment dynamics, and temporary immunity with waning. The use of the Caputo fractional derivative is motivated by the well-documented existence of chronic asymptomatic Salmonella Typhi carriers, whose heavy-tailed sojourn times in the carrier state are naturally encoded by the Mittag–Leffler waiting-time distribution arising from the fractional operator. A complete qualitative analysis of the fractional system is carried out: the basic reproduction number ${\mathcal{R}}_0$ is derived via the next-generation matrix method; local and global asymptotic stability of both the disease-free equilibrium ${\mathcal{E}}_0$ (when ${\mathcal{R}}_0\le 1$) and the endemic equilibrium ${\mathcal{E}}^{*}$ (when ${\mathcal{R}}_0>1$) are established using fractional Lyapunov theory and the LaSalle invariance principle; and the normalised sensitivity indices of ${\mathcal{R}}_0$ are computed to identify transmission-amplifying and transmission-suppressing parameters. Existence, uniqueness, and Ulam–Hyers stability of solutions are established via Banach and Leray–Schauder fixed-point arguments. To complement the analytical results, a fractional physics-informed neural network ($\mathcal{PINN}$) framework is developed to simultaneously reconstruct compartmental trajectories and identify unknown biological parameters from sparse synthetic observations. $\mathcal{PINN}$ embeds the L1-Caputo discretisation directly into the training residuals and employs a four-stage Adam–L-BFGS optimisation strategy to recover five trainable parameters $\mathbf{\Theta }$ = $\{\varphi ,\mu ,\sigma ,\psi ,\beta \}$ across three fractional orders $\kappa \in \{1.0,0.95,0.9\}$. The estimated parameters show strong agreement with the true values at the classical limit $\kappa =1.0$ ($\mathrm{MAPE}=2.27\%$), with the natural mortality rate $\mu$ recovered with $\mathrm{APE}\le 0.51\%$ and the transmission rate $\beta$ with $\mathrm{APE}\le 3.63\%$ across all fractional orders, confirming the structural identifiability of the model. Pairwise correlation analysis of the learned parameters establishes the absence of equifinality, validating that $\beta$ can be reliably included in the trainable set. Noise robustness experiments under Gaussian perturbations of $1\%$, $3\%$, and $5\%$ demonstrate graceful degradation ($\mathrm{MAPE}$: $0.82\%\to 3.10\%\to 7.31\%$), confirming the reliability of the proposed framework under realistic observational conditions. 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

Understanding how past factors influence ion channel kinetics is essential for understanding complex phenomena in cardiac electrophysiology, such as early afterdepolarizations (EADs), which are abnormal depolarizations during the action potential plateau associated with life-threatening arrhythmias. We developed a mathematical framework that extends Hodgkin-Huxley type equations with gamma Mittag-Leffler distributed delays, using tools from Fractional Calculus. Traditional memoryless two-variable models fail to reproduce EADs. Our approach modifies FitzHugh-Nagumo, Mitchell-Schaeffer, and Karma cardiac models, enabling the generation of EADs in each of them. We analyze the emergence of these oscillations by discussing the fractional parameters and the mean and variance of the memory kernels. Stability observations are also presented. Full article

Needed for cosmological and stellar nucleosynthesis, we are studying the closed-form analytic evaluation of thermonuclear reaction rates. In this context, we undertake a comprehensive analysis of three largely distinct velocity distributions, namely the Maxwell–Boltzmann distribution, the pathway distribution, and the Mittag-Leffler distribution. Moreover, a natural generalization of the Maxwell–Boltzmann velocity distribution is discussed. Furthermore, an explicit evaluation of the reaction rate integral in the high-energy cut-off case is carried out. Generalized special functions of mathematical physics like Meijer’s G-function and Fox’s H-functions and their utilization in mathematical physics are the prime focus of this paper. Full article

The L-fractional derivative is defined as a certain normalization of the well-known Caputo derivative, so alternative properties hold: smoothness and finite slope at the origin for the solution, velocity units for the vector field, and a differential form associated to the system. We develop a theory of this fractional derivative as follows. We prove a fundamental theorem of calculus. We deal with linear systems of autonomous homogeneous parts, which correspond to Caputo linear equations of non-autonomous homogeneous parts. The associated L-fractional integral operator, which is closely related to the beta function and the beta probability distribution, and the estimates for its norm in the Banach space of continuous functions play a key role in the development. The explicit solution is built by means of Picard’s iterations from a Mittag–Leffler-type function that mimics the standard exponential function. In the second part of the paper, we address autonomous linear equations of sequential type. We start with sequential order two and then move to arbitrary order by dealing with a power series. The classical theory of linear ordinary differential equations with constant coefficients is generalized, and we establish an analog of the method of undetermined coefficients. The last part of the paper is concerned with sequential linear equations of analytic coefficients and order two. Full article

This paper presents a fractional linear reservoir model as the elementary response function of hydrologic systems corresponding to the classical linear reservoir model and tests its applicability to rainfall–runoff modeling. To this end, we formulate a fractional linear reservoir model in terms of fractional calculus following the same procedure as the classical linear reservoir model and, at the simplest level, compare its performance of rainfall–runoff modeling with the linear and nonlinear reservoir models. The impulse response function of a fractional linear reservoir model, a probability density function (PDF) following the Mittag–Leffler distribution, shows nonlinearity due to its time-variant behavior compared to that of a linear reservoir model. In traditional linear hydrologic system theory, the lag and route version of a fractional linear reservoir model produces the fast-rising and slow-recession of runoff hydrographs, implying the mixed response of linear and nonlinear reservoir models to rainfall. So, a fractional linear reservoir model could be considered a fundamental tool to effectively reflect the nonlinearity of rainfall–runoff phenomena within the framework of the linear hydrologic system theory. In this respect, the fractional order of the storage relationship specifying a fractional linear reservoir model can be viewed as a kind of parameter to quantify the heterogeneity of runoff generation within a river basin. Full article

Very recently, a different generalization of real-valued neural networks (RVNNs) to multidimensional domains beside the complex-valued neural networks (CVNNs), quaternion-valued neural networks (QVNNs), and Clifford-valued neural networks (ClVNNs) has appeared, namely octonion-valued neural networks (OVNNs), which are not a subset of ClVNNs. They are defined on the octonion algebra, which is an 8D algebra over the reals, and is also the only other normed division algebra that can be defined over the reals beside the complex and quaternion algebras. On the other hand, fractional-order neural networks (FONNs) have also been very intensively researched in the recent past. Thus, the present work combines FONNs and OVNNs and puts forward a fractional-order octonion-valued neural network (FOOVNN) with neutral-type, time-varying, and distributed delays, a very general model not yet discussed in the literature, to our awareness. Sufficient criteria expressed as linear matrix inequalities (LMIs) and algebraic inequalities are deduced, which ensure the asymptotic and Mittag–Leffler synchronization properties of the proposed model by decomposing the OVNN system of equations into a real-valued one, in order to avoid the non-associativity problem of the octonion algebra. To accomplish synchronization, we use two different state feedback controllers, two different types of Lyapunov-like functionals in conjunction with two Halanay-type lemmas for FONNs, the free-weighting matrix method, a classical lemma, and Young’s inequality. The four theorems presented in the paper are each illustrated by a numerical example. Full article

Hyperbolic complete monotonicity property ($\mathrm{HCM}$) is a way to check if a distribution is a generalized gamma ($\mathrm{GGC}$), hence is infinitely divisible. In this work, we illustrate to which extent the Mittag-Leffler functions $E_{\alpha },\alpha \in (0,2]$, enjoy the $\mathrm{HCM}$ property, and then intervene deeply in the probabilistic context. We prove that for suitable $\alpha$ and complex numbers z, the real and imaginary part of the functions $x\mapsto E_{\alpha }\left(zx\right)$, are tightly linked to the stable distributions and to the generalized Cauchy kernel. Full article

A class of generalized fractional Zener-type viscoelastic models with general fractional derivatives is considered. Two integral representations are derived for the corresponding relaxation modulus. The first representation is established by applying the Laplace transform to the constitutive equation and using the Bernstein functions technique to justify the change of integration contour in the complex Laplace inversion formula. The second integral representation for the relaxation modulus is obtained by applying the subordination principle for the relaxation equation with generalized fractional derivatives. Two particular examples of the considered class of models are discussed in more detail: a model with fractional derivatives of uniformly distributed order and a model with general fractional derivatives, the kernel of which is a multinomial Mittag-Leffler-type function. To illustrate the analytical results, some numerical examples are presented. Full article

A review of the literature demonstrates that the Fox–Wright function is not only a mathematical puzzle, but its role is naturally to represent basic physical phenomena. Motivated by this fact, we studied a new representation of this function in terms of complex delta functions. This representation was useful to compute its Laplace transform with respect to the third parameter γ for which it also generalizes the one and two-parameter Mittag-Leffler functions. New identities involving the Fox–Wright function were discussed and used to simplify the results. Different fractional transforms were evaluated and the solution of a fractional kinetic equation was obtained by using its new representation. Several new properties of this function were discussed as a distribution. Full article

A non-Markovian model of tumor cell invasion with finite velocity is proposed to describe the proliferation and migration dichotomy of cancer cells. The model considers transitions with age-dependent switching rates between three states: moving tumor cells in the positive direction, moving tumor cells in the negative direction, and resting tumor cells. The first two states correspond to a migratory phenotype, while the third state represents a proliferative phenotype. Proliferation is modeled using a logistic growth equation. The transport of tumor cells is described by a persistent random walk with general residence time distributions. The nonlinear master equations describing the average densities of cancer cells for each of the three states are derived. The present work also includes the analysis of models involving power law distributed random time, highlighting the dominance of the Mittag–Leffler rest state, resulting in subdiffusive behavior. Full article

Sun is a basic component of the natural environment and kinetic equations are important mathematical models to assess the rate of change of chemical composition of a star such as the sun. In this article, a new fractional kinetic equation is formulated and solved using generalized Krätzel integrals because the nuclear reaction rate in astrophysics is represented in terms of these integrals. Furthermore, new identities involving Fox–Wright function are discussed and used to simplify the results. We compute new fractional calculus formulae involving the Krätzel function by using Kiryakova’s fractional integral and derivative operators which led to several new identities for a variety of other classic fractional transforms. A number of new identities for the generalized Krätzel function are then analyzed in relation to the $H$-function. The closed form of such results is also expressible in terms of Mittag-Leffler function. Distributional representation of Krätzel function and its Laplace transform has been essential in achieving the goals of this work. Full article