Derivative Operator Research Articles

Analytical insights into a fractional thin-film equation: exact solutions and dynamics

In this work, we examine a quadratic thin film equation with a constant negative absorption term. This equation extends a broad variety of the famous scalar reaction-diffusion equations appearing in nonlinear sciences and is derived from the estimations of lubrication theory to represent thin films of a Newtonian liquid dominated by surface tension effects. It is typically used to describe the behavior of light when it interacts with thin films, such as coatings on lenses or mirrors. The connection between thin film equations and optical quantum mechanics lies in the microscopic interactions between photons and the electrons in the thin film material. Employing the invariant subspace approach, we obtain explicit fractional exact solutions for the time-fractional case of the model containing the Riemann-Liouville derivative operator. Furthermore, we illustrate 3-D and 2-D plots of the obtained exact solutions for a better understanding of the physical phenomena.

A universal formula for derivation operators and applications

We give a universal formula describing derivation operators on a Hilbert space for a large class of interpolation methods. It is based on a simple new technique on “critical points” where all the derivations attain the maximum. We deduce from this a version of Kalton uniqueness theorem for such methods, in particular, for the real method. As an application of our ideas is the construction of a weak Hilbert space induced by the real J-method. Previously, such space was only known arising from the complex method. To complete the picture, we show, using a breakthrough of Johnson and Szankowski, nontrivial derivations whose values on the critical points grow to infinity as slowly as we wish.

Necessity of quantizable geometry for quantum gravity

In this paper, Dirac Quantization of 3D gravity in the first-order formalism is attempted where instead of quantizing the connection and triad fields, the connection and the triad 1-forms themselves are quantized. The exterior derivative operator on the space of differential forms is treated as the ‘time’ derivative to compute the momenta conjugate to these 1-forms. This manner of quantization allows one to compute the transition amplitude in 3D gravity which has a close, but not exact, match with the transition amplitude computed via LQG techniques. This inconsistency is interpreted as being due to the non-quantizable nature of differential geometry.

A geometric framework for distributed frequency models

Geometric control theory, developed by Basile and Marro, and independently, by Wonham and Morse in the 1970s revolves around characterizing the properties of finite dimensional, linear and time-invariant systems using geometry. Some examples of these properties are invariance, controllability and observability. The task addressed in this paper is to develop the geometric tools for fractional systems using the diffusive representation (also known as the distributed frequency) model. The mathematics involved in this approach is different from the classical case as fractional systems are inherently infinite dimensional. Unlike the integer order derivative, the fractional derivative is a non-local operator, and so the evolution of the so-called pseudo-state depends on not just its current value but also its past history. Thus, the notion of an initial condition for a fractional system can come in different forms. This leads to different kinds of fractional derivative operators. With some of these operators, the semigroup property is lost. With the distributed frequency model, the initial condition comes in the form of an initialization function. This takes care of the infinite dimensional nature of fractional systems. Furthermore, the distributed frequency model retains the semigroup property. This is useful in developing invariance and controlled invariance for fractional systems. Moreover, these properties of fractional systems are verified numerically using a higher order finite-dimensional approximation, which retains all the geometric properties of the distributed frequency model.

On Certain Properties of Parametric Kinds of Apostol-Type Frobenius–Euler–Fibonacci Polynomials

This paper presents an overview of cosine and sine Apostol-type Frobenius–Euler–Fibonacci polynomials, as well as several identities that are associated with these polynomials. By applying a partial derivative operator to the generating functions, the authors obtain derivative formulae and finite combinatorial sums involving these polynomials and numbers. Additionally, the paper establishes connections between cosine and sine Apostol-type Frobenius–Euler–Fibonacci polynomials of order α and several other polynomial sequences, such as the Apostol-type Bernoulli–Fibonacci polynomials, the Apostol-type Euler–Fibonacci polynomials, the Apostol-type Genocchi–Fibonacci polynomials, and the Stirling–Fibonacci numbers of the second kind. The authors also provide computational formulae and graphical representations of these polynomials using the Mathematica program.

A new generalized beta function associated with statistical distribution and fractional kinetic equation

Several authors have extensively investigated beta function, hypergeometric function and confluent hypergeometric function, their extensions and generalizations due to their several application in many areas of engineering, probability theory and science. The main purpose of this paper is to present a new generalization of extended beta function, hypergeometric function and confluent hypergeometric function with the help of m-parameter Mittag-Leffler function, as well as examine some important properties like integral representations, differential formula and summation formulas. We also examine generalized Caputo fractional derivative operator with the help of m-parameter Mittag-Leffler function with associated properties using the generalized beta function. We define a new beta distribution involving the new generalized beta function. The mean, variance, coefficient of variance, moment generating function and characteristic function and cumulative distribution are derived. Further, we derive the solution of fractional Kinetic equation involving generalized hypergeometric function.

Novel Numerical Solutions to the Fractional-Stochastic Systems

Fractional-stochastic differential equations are widely used tools to simulate a wide range of engineering and scientific phenomena. In this paper, the applicability of the approach of indeterminate coefficients to various fractional-stochastic models is examined. These models have a fractional white noise term and are mostly produced by fractional-order derivative operators. We also look into the application of a polynomial chaos algorithm to stochastic Lotka-Volterra and Benney systems. Fractional-stochastic equations are entirely novel systems that have the potential to function as models for a wide range of scientific and engineering phenomena. It is noted that fractional-order systems with uncertainty or a noise term can benefit from the effective use of Galerkin type approaches used in this article. Keywords: Galerkin method, Numerical solutions, Stochastic-fractional systems

Validating the feasibility of the method of ensemble learning combined with FT-MIR for the discrimination of wild Paris polyphylla var. Yunnanensis from different geographical sources

As a widely used medicinal plant with high economic value, Paris polyphylla var. yunnanensis (PPY), is mainly distributed in south-western China. Since different growing environments have a strong influence on the PPY quality, efficient means are needed to discriminate its origin source. The traditional machine learning model uses a single learner to classify and discriminate the samples, but it is difficult to apply its advantages to play for complex data samples. Ensemble learning can integrate the advantages of multiple learners to improve the model performance. In this experiment, Partial Least Squares Discriminant Analysis (PLS-DA), Random Subspace Method (RSM), Random Forest (RF), and Support Vector Machine (SVM) models were used to discriminate wild PPY from different origins, and then verify the feasibility of ensemble learning in discriminating them from different geographical sources. At the same time, the results of preprocessing the spectral data with multiplicative scattering correction (MSC), standard normal variate (SNV), and derivative operations (first, second and third derivatives) and their combination methods were compared. The results show that the combined preprocessing greatly improves the model performance. The RSM has the most stable performance with an accuracy of more than 85%. Indicating that the ensemble learning combined with Fourier transform mid-infrared spectroscopy (FT-MIR) has the feasibility of discriminating the wild PPY of different geographical origins.

Some Applications of Fractional Derivative Operator

Introduction: The aim of this paper is to introduce a new subclass of univalent functions with negative coefficients related to fractional derivative operator in the unit disk We obtain basic properties like coefficient inequality, distortion and covering theorem, radii of starlikeness, convexity and close-to-convexity, extreme points, Hadamard product, and closure theorems for functions belonging to our class

Finite Difference Method for Infection Model of HPV with Cervical Cancer under Caputo Operator

In this paper, a fractional model in the Caputo sense is used to characterize the dynamics of HPV with cervical cancer. Generalized mean value theorem has been used to examine whether the infection model has a unique positive solution. The model has two equilibrium points: the disease-free point and the endemic point. The examination of the system’s local and global stability is provided in terms of the basic reproductive number Rp°. The global stability analysis has been carried out using an appropriate Lyapunov function and the LaSalle invariant principle. The results demonstrate that in the infection model, if Rp°<1, then the solution converges to the disease-free equilibrium, which is both locally and globally asymptotically stable. Whilst Rp°>1, the endemic equilibrium is considered to exist. Simulations are implemented via a finite difference method with Grünwald-Letnikov discretization approach for Caputo derivative operator to define how changes in parameters impact the dynamic behavior of the system using Matlab.

Higher order quantum waves in fractal dimensions from nonlocal complex derivative operator

In this study, we introduced a new nonlocal complex derivative operator in fractal dimension based concurrently on the concept of “nonlocal generalized complex backward-forward coordinates” and the “product-like fractal measure”. The quantization of the theory in fractal dimension leads to a higher order Schrödinger equation characterized by a higher order energy operator. As an illustration, we have discussed the cases of infinite quantum well and power-law potentials. Their associated zero-point energies were found to depend on the numerical value of the fractal dimension. For the infinite well, the decrease in zero-point energy with fractal dimension may result in the emission of large wavelengths photons observed experimentally in X-ray laser bursts emitted from the solid.

De-projecting the EFThedron

The space of Wilson coefficients of EFT that can be UV completed into consistent theories was recently shown to be described analytically by a positive geometry, termed the EFThedron. However, this geometry, as well as complementary numerical methods of semi-definite programming, have so far focused on the positivity of the partial wave expansion, which allows bounding only ratios of couplings. In this paper we describe how the unitarity upper bound of the partial waves can be incorporated. This new problem can be formulated in terms of the well known L-moment problem, which we generalize and solve from a geometrical perspective. We find the non-projective generalization of the EFThedron has an infinite number of non-linear facets, which in some cases have remarkably simple descriptions. We use these results to derive bounds on single couplings, finding that the leading derivative operators are bounded by unity, when normalized by the cut-off scale and loop factors. For general operators of mass dimension 2k we find the upper bound is heavily suppressed at large k, with an 1/k fall-off.

P -wave fully charm and fully bottom tetraquark states

We have studied the mass spectra of the P-wave fully charm and fully bottom tetraquark states in the framework of quantum chromodynamics (QCD) sum rules. We construct the interpolating currents by inserting the covariant derivative operator D↔μ between the S-wave diquark and antidiquark fields. The excitation structures show that the pure λ-mode excited P-wave fully heavy tetraquarks exist for the quantum numbers JPC=1−−,1−+,2−−,2−+, and 3−−, while it is difficult to separate the λ-mode and ρ-mode excitations in the 0−+ channel. Within three Lorentz indices, there is no pure λ-mode excited P-wave fully charm/bottom tetraquark operators with JPC=0−− and 3−+. Our results support that the recent observed X(6900) and X(7200) resonances could be interpreted as the P-wave fully charm ccc¯c¯ tetraquark states with JPC=1−+ and 2−+, respectively. Some P-wave fully bottom bbb¯b¯ tetraquark states are predicted to be lower than the di-ηb(1S) and di-ϒ(1S) mass thresholds. Hopefully our calculations will be useful for identifying the nature of new exotic tetraquark states. Published by the American Physical Society 2024

A powerful tool for dealing with high-dimensional fractional-order systems with applications to fractional Emden–Fowler systems

In this study, an approximation solution for a high-dimensional system in terms of the Caputo fractional derivative operator is obtained using the improved modified fractional Euler method, or IMFEM for short. To accomplish this aim, a result that can transform such a system into a double-equation, one-dimensional fractional-order system, is provided theoretically. Some physical applications, including fractional-order systems of equations of Emden–Fowler type, are discussed, and their graphs are plotted using MATLAB to demonstrate the IMFEM schema’s validity.

Mixed Hilfer and Caputo Fractional Riemann–Stieltjes Integro-Differential Equations with Non-Separated Boundary Conditions

In this paper, we investigate a sequential fractional boundary value problem which contains a combination of Hilfer and Caputo fractional derivative operators and non-separated boundary conditions. We establish the existence of a unique solution via Banach’s fixed point theorem, while by applying Leray–Schauder’s nonlinear alternative, we prove an existence result. Finally, examples are provided to demonstrate the results obtained.

An Extended Subclass of Meromorphic Multivalent Functions Involving Ruscheweyh Derivative Operator

In this paper, we introduce and discuss an extended subclass〖 Ą〗_p^*(λ,α,γ) of meromorphic multivalent functions involving Ruscheweyh derivative operator. Coefficients inequality, distortion theorems, closure theorem for this subclass are obtained.

A meshfree method for the nonlinear KdV equation using stabilized collocation method and gradient reproducing kernel approximations

A gradient reproducing kernel based stabilized collocation method (GRKSCM) to numerically solve complicated nonlinear Korteweg-de Vries (KdV) equation is proposed in this paper. The acquisition of GRK through high-order consistency conditions reduces the complexity of RK derivative operations and remarkably improves the effectiveness of the proposed method by directly forming GRK approximations. Owing to the fulfillment of high-order integration constraints, the SCM achieves exact integration in the domain and on boundaries. Von Neumann analysis is utilized to establish the stability criteria for GRKSCM when combined with forward difference temporal discretization. The effectiveness of the GRKSCM method in solving the KdV equation is investigated through six numerical examples. The examples include the motion of the single solitary wave propagation, interaction between two solitary waves and interaction among three solitary waves. Furthermore, the behavior of a two-dimensional solitary wave and the propagation of a three-dimensional solitary wave are also numerically investigated. The numerical outcomes confirm that GRKSCM provides high accuracy in comparison with analytical solutions. In addition, the invariants and error analysis show the conservation of our proposed GRKSCM method.

Mathematical analysis of a within-host dengue virus dynamics model with adaptive immunity using Caputo fractional-order derivatives

AbstractDengue fever poses a significant global health threat, with over 50 million annual infections spanning more than 100 countries. Given the absence of a specific treatment, medical intervention primarily targets symptom alleviation. The present study utilizes a Caputo-type fractional-order derivative operator to investigate and analyze the dynamics of dengue virus spread within a host with adaptive immune responses. The developed model describes and analyzes the dynamics of immune cells, free dengue particles, infected monocytes, and susceptible monocytes in the presence of cytotoxic T-Lymphocytes. A range of analytical methods is employed to probe the fractional-order within-host model. The application of the generalized mean value theorem aids in investigating the model’s solutions, employing positivity and boundedness theory. Furthermore, the Banach fixed-point approach is utilized to establish the existence and uniqueness of solutions. Employing the normalized forward sensitivity approach, the fractional-order system’s response to various model parameters is scrutinized. The study reveals that the dynamics of the viral model are significantly influenced by the transmission rate and parameters representing adaptive immune responses. Numerical simulations underscore the critical role of transmission rates and adaptive immune responses in the model. Additionally, the study examines the impact of memory on the density of susceptible monocytes, infected monocytes, free dengue particles, and immune cells to optimize immune responses. Through simulations, the study illustrates the influence of memory on immune dynamics.

FRACTAL FRACTIONAL MODEL OF RADIATIVE HEAT AND MASS TRANSFER CHARACTERISTICS OF TIME DEPENDENT FLOW WITH VARIABLE VISCOSITY, INDUCED MAGNETIC FIELD, DUFOUR AND SORET EFFECTS: AN ENTROPY GENERATION

: In this article, the Caputo Fabrizio fractal fractional order derivatives operator with an exponential kernel was employed to examine the radiative heat and mass transfer characteristics of time dependent flow with variable fluid viscosity, induced magnetic field, Soret and Dufour effects. The local mathematical model for the flow problem is formulated by take into account the impacts of thermal radiation, heat source, and viscous dissipation. The governing equations in-terms of fractal fractional model with an exponential kernel were solved numerically using finite difference method. The influence of flow variables such as induced magnetic field, concentration field, entropy rate, thermal field, and velocity field profiles against the pertinent parameters are discussed through graphs. Increase the values of magnetic Prandtl number results to rises of induced magnetic field. Higher Dufour number significantly grows the thermal field. The fractal fractional parameters enhance the velocity field, thermal field, Bejan number, entropy rate, concentration field and induced magnetic field profiles. The velocity field profiles recede with higher values of fluid variable viscosity parameter whereas the thermal field and induced magnetic field has an opposite effect. Larger Soret number amplifies the concentration field. Increase of Brinkman number, thermal radiation parameter, and magnetic Prandtl number intensifies the entropy generation rate. Increases of Brinkman number, magnetic Prandtl number, Soret number and Dufour number leads to a decrease of Bejan number whereas Bejan number rises with an increase of thermal radiation parameter and heat source

The modified fractional‐order quasi‐reversibility method for a class of direct and inverse problems governed by time‐fractional heat equations with involution perturbation

This study aims to explore two classes of ill‐posed problems governed by a nonclassical fractional heat equation with an involution perturbation. To achieve a stable solution, we introduce a modified pseudo‐parabolic regularization method that involves a correction term through a mixed fractional derivation operator, resulting in a sequence of well‐posed problems that depend on a regularization parameter . We rigorously demonstrate that the resulting approximate problems are well‐posed and present convergence results for this regularization approach.