General fractional integral operators do not preserve periodicity

A novel stochastic fractal search operator based on particle swarm optimization for constrained multi-objective optimization

ABSTRACT Leveraging the Liouville–Caputo fractional derivative (LCFD), this work constructs a mathematical framework to examine the dynamics of Human Papillomavirus (HPV) transmission and its progression into cervical cancer. We propose a novel fractional‐order model (FOM) consisting of five coupled fractional differential equations (FDEs) that represent different population compartments and incorporate essential epidemiological characteristics of HPV infection. To ensure mathematical validity, we rigorously prove that the model's solutions exist, are unique, and remain positive and bounded. Our analysis involves deriving the basic reproduction number to establish a threshold for disease persistence, followed by a comprehensive examination of the stability of the disease‐free and endemic equilibrium states. Parameter sensitivity analysis identifies key factors influencing disease transmission dynamics. An extension of our framework involves the formulation of a fractional optimal control problem (FOCP), which includes three time‐dependent control measures: safe sexual practice promotion (), vaccination and immunity enhancement (), and comprehensive cancer screening and treatment implementation (). By employing Pontryagin's Maximum Principle (PMP), we rigorously derive the set of necessary conditions required for optimal control. To numerically solve and validate the model, a deep neural network (DNN) with Tanh and ReLU activations was implemented within a stochastic framework. The model's fidelity was verified through convergence testing, error distribution analysis, and regression metrics, using the Adams‐Bashforth‐Moulton (ABM) scheme and with data divided for training (70%), validation (15%), and testing (15%). The proposed methodology yields solutions that align precisely with benchmark data, achieving a best validation performance of and a minimum absolute error of . Numerical simulations of four control strategies demonstrate that while each intervention reduces infection and cancer progression to some extent, the combined strategy proves most effective. This study highlights the pivotal role of the fractional operator in disease dynamics and introduces a novel hybrid approach that integrates deep learning with fractional calculus, offering a computationally efficient and highly accurate tool for analyzing complex epidemiological systems.

This paper addresses the existence of mild solutions and the approximate controllability of a class of higher-order Hilfer fractional semi-linear neutral stochastic differential equations with non-instantaneous impulses in Hilbert spaces. The system is driven by both fractional Brownian motion and Poisson jumps, thereby capturing long-range dependence as well as random discontinuities. By combining techniques from fractional calculus, stochastic analysis, and operator theory, we establish sufficient conditions for the existence of mild solutions. The analysis is carried out through the construction of suitable solution operator families and the application of Sadovskii’s fixed point theorem in an appropriate phase space framework. In addition, we investigate the controllability properties of the system and derive criteria ensuring approximate controllability of the underlying fractional neutral dynamics. The proposed approach relies on the structural properties of the higher-order Hilfer fractional derivative, estimates for stochastic integrals with respect to fractional Brownian motion, and compactness arguments adapted to non-instantaneous impulsive effects. The inclusion of Poisson jumps and neutral terms introduces significant analytical difficulties, which are overcome using refined resolvent operator techniques and fractional power estimates. An illustrative example is presented to demonstrate the applicability of the theoretical results. The results obtained generalize and unify several recent developments in the theory of fractional stochastic systems and provide a flexible framework for analyzing controlled dynamical models with memory, randomness, and impulsive behavior.

Abstract Tunneling ionization in static or slowly varying electric fields is a cornerstone of strong-field physics and provides the entry point for semiclassical descriptions of above-threshold ionization and high-harmonic generation. In conventional quantum mechanics, the Perelomov-Popov-Terent'ev (PPT) theory and its Ammosov-Delone-Krainov (ADK) form yield an ionization rate whose defining feature is an exponential dependence governed by an under-barrier (imaginary-time) action. Here we develop an analytical ADK-like tunneling model within space-fractional quantum mechanics, where the quadratic kinetic energy is replaced by the Riesz fractional Laplacian of order 1 < α ≤ 2. Working in a static electric field in the length gauge, we derive a closed-form tunneling exponent for a triangular exit barrier. The fractional kinetic operator deforms the conventional I 3/2 p scaling to I 1+1/α p and introduces a characteristic sin(π/α) factor encoding the complex-phase structure associated with nonlocal dispersion. We position this benchmark relative to prior tunneling studies in fractional quantum mechanics (primarily scattering through model barriers and fractal potentials) and provide a validation protocol for testing the exponent in time-dependent simulations of the fractional Schrödinger equation under a constant field. The result establishes a transparent reference for static-field ionization in nonlocal quantum dynamics and a baseline for strong-field approaches extensions.

The main concern of this work is to study the sufficient conditions for existence, uniqueness, and approximate controllability results for the nonlinear Caputo conformable fractional neutral-type delayed integro-differential system with nonlocal conditions in a Hilbert space. Since, the conformable derivative retains several fundamental properties of classical calculus including mean value theorem, Rolle's theorem, product, quotient, and linearity rules. This distinguishes it from traditional fractional derivatives such as Riemann-Liouville, Caputo, and Hilfer. Therefore, the conformable derivative is simpler and faster but ignores history while the Caputo conformable fractional derivative offers a balance capturing some memory with easier calculations. Firstly, the proposed system is reformulated into an equivalent fixed point problem implementing the Riemann-Liouville conformable fractional integral operator. The Schauder fixed point theorem is used to derive the existence of mild solution. The Banach contraction principle is then applied to show the uniqueness of mild solution. The main tools applied to derive the results are theory of fractional calculus, semigroup of bounded linear operators, and fixed point theorems. Further, the approximate controllability result for the proposed system is established under the consideration that the corresponding linear system is approximate controllable. An illustrative example is presented to demonstrate the applicability of theoretical results.

The Le Roy function has been the focus of intensive research in recent years, owing both to its relevance in analysis and its versatility in applications involving fractional differential operators. Other special functions – such as the Lerch transcendent and the Legendre chi function – have found applications ranging from Bose-Einstein and Fermi-Dirac statistics in physics to pure mathematical investigations involving polylogarithms and Dirichlet L-series. In this article, we present a unified framework based on a recent reformulation of Indicial Umbral Theory (IUT) grounded in the formal theory of power series. Within this setting, we study the properties and generalizations of these special functions. In particular, we build upon the revised formulation of IUT to incorporate the role of the Borel-Le Roy transform, and to explore the extension of the formalism to divergent series via appropriate resummation techniques.

This paper presents an innovative parallel computational methodology employing adaptive fractional differential operators to address Monkeypox epidemiological modeling challenges. The complex dynamics of such models often present computational challenges. To address these, we develop a productive hybrid graphics processing unit–central processing unit (GPU–CPU) parallel approach. Using fractional differentiation operators, the presented approach focuses on managing the memory component of disease transmission dynamics. Our parallel method significantly reduces the calculation time and improves the overall performance of solving the Monkeypox disease model by utilizing the computing capabilities of both GPU and CPU cores. In this paper, we provide a hybrid variable-order fractional derivative that combines the variable-order fractional Caputo derivative with the integral of Riemann-Liouville. A predictor-corrector method with discretization of the Caputo proportional constant variable-order fractional hybrid operator is applied for numerical solutions. Julia, a high-level programming language, was chosen to implement the hybrid parallel technique. According to the results of the simulation, parallel approaches significantly increase productivity and efficiency.

Abstract This study investigates the existence and uniqueness of solutions for a coupled system of Langevin-type fractional differential equations featuring generalized Ψ-Caputo derivatives in arbitrary Banach spaces. While prior research has predominantly focused on finite-dimensional or specific function spaces, this work extends the framework to infinite-dimensional settings, offering a more versatile analytical approach. Uniqueness is established using Banach’s fixed-point theorem under Lipschitz-type conditions, while existence is proven via Monch’s fixed-point principle combined with the measure of noncompactness—a powerful tool for infinite-dimensional problems. Demonstrative examples, including cases in the space of null sequences, validate the theoretical framework. The findings enhance the study of fractional coupled systems, introducing a flexible and comprehensive approach that integrates diverse fractional operators and strengthens foundational results.

Modeling the propagation of fractional inequalities via generalized fractional operators with artificial neural networks

In this work, a time-fractional form of Richards' equation is considered to study the infiltration phenomenon in unsaturated porous media. The memory effects in the flow process are described using the Caputo-Fabrizio fractional operator. To obtain an approximate analytical solution of the governing nonlinear fractional partial differential equation, a hybrid analytical technique combining the Natural transform method and the variational iteration method is employed. The proposed Natural transform variational iteration method (NVIM) provides a rapidly convergent series solution and avoids complicated discretization or linearization procedures. A rigorous convergence and uniqueness analysis is carried out, which confirms the validity and accuracy of the proposed approach. The effectiveness and reliability of the method are demonstrated through the solution of the considered problem. The obtained solutions are illustrated through graphical representations using MATHEMATICA, where a comparison between the approximate analytical solution and the exact solution is carried out to validate the accuracy and effectiveness of the proposed method. In addition, the influence of the fractional parameter and other model parameters on the infiltration process is analyzed through graphical illustrations. The results demonstrate that the proposed approach provides accurate and efficient solutions and can serve as a useful analytical tool for solving nonlinear fractional differential equations arising in groundwater hydrology and related physical processes.

Numerical reconstruction of the kernel function in generalized non-convolutional fractional operators

New adaptive memory stochastic fractional operators and their applications in computational intelligence.

We study fractional and complex powers of a fixed directional derivative in Rd, defined via a Marchaud-type singular integral representation. Under explicit convergence assumptions, this yields a pointwise nonlocal realization along rays. We then formulate a Ramanujan–Hardy approach to fractional directional differentiation based on analytic interpolation of the directional jet at a point. This construction is local in jet space and is governed by Hardy’s formulation of Ramanujan’s Master Theorem. We emphasize that the resulting Ramanujan–Hardy derivative is defined through a Hardy-admissible interpolant of the directional jet. As an application, we investigate fractional directional derivatives of the Newtonian kernel in dimension d≥3. After a justified regularization and reduction to a Marchaud-type integral, we obtain a one-dimensional integral representation and a zonal harmonic description of the resulting function. This leads to a fractional Maxwell–Gegenbauer identity for 0<ℜ(s)<1, expressing the fractional directional derivative of ∥x∥2−d in terms of Gegenbauer functions of complex degree. In this way, the classical Maxwell multipole formula appears as the integer-order case of a continuous analytic family. Moreover, the fractional operator preserves the main structural properties of the Newtonian kernel, including homogeneity, rotational invariance, and harmonicity away from the origin. The paper thus connects Mellin analysis, Ramanujan’s Master Theorem, fractional calculus, and harmonic analysis on the sphere, while clarifying the distinction between Marchaud and jet-interpolation constructions of fractional directional operators.

In this paper, we give some new Jensen, Jensen–Mercer, and Hermite–Hadamard inequalities for uniformly δ-geometric convex functions. In addition, some limit bounds for Caputo–Fabrizio fractional integral operators are established as an application in the case of uniformly δ-geometric convex functions. Some new examples and graphical representations are provided in order to illustrate the validity of our results.

Fractional calculus modifies the geometric roughness of functions and alters global fractal dimensions in a monotonic manner. However, a comprehensive description of how fractional operators transform full scaling structures and multifractal spectra remains incomplete. In this paper, we establish a unified scaling-law theory describing the action of Riemann- Liouville and Weyl-Marchaud fractional operators on local regularity, structure functions, and multifractal spectra. We prove that fractional differentiation of order [Formula: see text] induces a translation of local Hölder exponents by [Formula: see text], yielding an exact shift law for the multifractal spectrum. Furthermore, we derive an affine transformation rule for structure-function scaling exponents, showing that fractional operators generate linear deformations of scaling laws across statistical moments. An AI-assisted identification framework is introduced to recover fractional order directly from observed fractal signals. Numerical experiments on synthetic multifractal processes confirm the theoretical predictions and demonstrate the robustness of the proposed method. These results provide a geometric and scaling interpretation of fractional calculus and establish a unified framework connecting fractal geometry, scaling laws, and artificial intelligence.

This paper develops a unified framework of fractional product inequalities for special functions, extending classical Cauchy-Schwarz type results to nonclassical convexity settings. The approach exploits the structural properties of generalized harmonic [Formula: see text]-convex functions in combination with local fractional integral operators to derive sharp and explicitly computable bounds. The analysis employs auxiliary identities together with refined Hölder and power-mean techniques adapted to the fractional context. Within this framework, both direct and reverse fractional product inequalities are established, with the deformation mechanisms and coefficient functions characterized in terms of fractional order and convexity parameters. Applications are presented for several important classes of special functions, including Gamma-type, Bessel-type, and hypergeometric-type functions. Moreover, many existing inequalities are recovered as particular or limiting cases, demonstrating the flexibility and generality of the approach. These results provide a broad and practical extension of classical inequality techniques, with potential relevance to fractional calculus, analysis, and related mathematical applications.

Fractional derivatives have demonstrated a wide range of solutions beneficial for engineering, medical, and manufacturing sciences. Research on the use of fractional derivatives in fluid flow problems is still emerging, especially in analytical studies. In this paper, a mathematical model is developed to analyze the behavior of Casson fluid in heat and mass transfer processes within a magnetohydrodynamic (MHD) environment over an unsteady oscillating surface. Additionally, the numerical computation of fractional MHD Casson fluid flow for parallel surfaces subject to oscillating boundary conditions is discussed. The model utilizes a non-singular fractional derivative to solve fractional-order partial differential equations. By applying an appropriate transformation, these equations are converted into a dimensionless form. Subsequently, the finite difference method is employed to analyze the system’s behavior. A comprehensive parametric study is conducted to explore the effects of different physical parameters on the flow and thermal fields, and numerical simulations identify the fractional derivative operator that yields the most accurate and stable solutions. Furthermore, the model is solved using an ANN approach, and multiple cases are discussed for varying values of the fractional order parameter [Formula: see text]. This ANNbased analysis confirms the effectiveness of the fractional model in capturing the complex dynamics of MHD Casson fluid flow under oscillating boundary conditions.

We develop a unified mathematical and computational framework linking fractional calculus, fractal geometry, scaling laws, and artificial intelligence. We establish a universal scaling-shift theorem demonstrating that fractional operators systematically transform fractal and multifractal spectra of self-affine stochastic processes. For any self-affine process with Hurst exponent [Formula: see text], we prove that a fractional derivative of order [Formula: see text] produces a new process with scaling exponent [Formula: see text], implying a fractal dimension shift [Formula: see text] We further derive a multifractal spectral deformation law [Formula: see text] which demonstrate how fractional dynamics rearranges local singularities. To connect theory with observations, we present an AI-facilitated framework of neural operators that can compute fractal dimensions and detect fractional orders from observations. Numerical experiments confirm the theoretical predictions and demonstrate strong agreement between analytical scaling laws and AI-inferred parameters.

We start with a random polynomial P N ( z ) P^{N}(z) of degree N N with independent coefficients. We then consider a new polynomial P t N P_{t}^{N} obtained by ⌈ N t ⌉ \lceil Nt\rceil applications of a fractional differential operator of the form z a ( d / d z ) b z^{a} (d/dz)^{b} , where a a and b b are real numbers. When b > 0 b>0 , we compute the limiting root distribution μ t \mu _{t} of P t N P_{t}^{N} as N → ∞ N\rightarrow \infty . We show that μ t \mu _{t} is the push-forward of the limiting root distribution of P N P^{N} under a transport map T t T_{t} . The map T t T_{t} is defined by flowing along the characteristic curves of a partial differential equation satisfied by the log potential of μ t \mu _{t} . In the special case of repeated differentiation, our results may be interpreted as saying that the roots evolve radially with constant speed until they hit the origin, at which point, they cease to exist. For general a a and b b , the transport map T t T_{t} has a free probability interpretation as multiplication of an R R -diagonal operator by an R R -diagonal “transport operator.” As an application, we obtain a push-forward characterization of the free self-convolution semigroup ⊕ \oplus of radial measures on C \mathbb {C} . We also consider the case b > 0 b>0 , which includes the case of repeated integration. More complicated behavior of the roots can occur in this case.