Derivative Operator Research Articles

2D Discrete Yang–Mills Equations on the Torus

In this paper, we introduce a discretization scheme for the Yang–Mills equations in the two-dimensional case using a framework based on discrete exterior calculus. Within this framework, we define discrete versions of the exterior covariant derivative operator and its adjoint, which capture essential geometric features similar to their continuous counterparts. Our focus is on discrete models defined on a combinatorial torus, where the discrete Yang–Mills equations are presented in the form of both a system of difference equations and a matrix form.

On a fractional operator of adjoint hybrid fractional derivative operator

The achievement of this paper is to propose a new kind of fractional derivative which is called New Constant Proportional Caputo (NCPC) operator and to construct the solution of time-fractional initial value problem (TFIVPs) with NCPC derivative by taking the combination of Laplace transform (LT) and Homotopy Analysis method (HAM) into account. Later, the obtained solution is compared with the solutions of TFIVPs with Caputo and Constant Proportional Caputo (CPC) derivatives. The gained results reveal that the combination of LT and HAM together form an efficient method to build the approximate results of TFIVPs in NCPC sense.

Darboux and generalized Darboux transformations for the fractional integrable derivative nonlinear Schrödinger equation

Analytical methods provide crucial mathematical insights into the stable solitary waves hidden in nonlinear phenomena. The nonlinear Schrödinger (NLS) equation is one of the most important typical integrable soliton models. From a mathematical perspective, the essence of the celebrated derivative NLS (DNLS) equation’s difference from the classical NLS equation lies in its cubic potential being differentiated once by the spatial variable and multiplied by the imaginary unit, which leads to the former having some characteristics that the latter cannot have. This paper extends the DNLS equation to the fractional integrable case with conformable derivative operators, and uses Darboux transformations (DTs) and generalized DT (GDT) to solve it exactly. Specifically, Lax pairs generating the fractional DNLS equation are first given. Based on the given Lax pairs, then the n-fold DTs and GDT for the fractional DNLS equation are derived. Some special exact solutions of the fractional DNLS equation are further obtained by employing the derived n-fold DTs and GDT. Finally, several novel space–time structures and dynamical evolutions of the obtained exact solutions are analyzed. This paper reveals through the DT and GDT methods that the double power-law fractional orders in the exact solutions of the fractional DNLS equation can be used to dominate the variable velocity propagation and anomalous diffusion in fractional dimensional media at different geometric scales.

A new fractional derivative operator with a generalized exponential kernel

A new fractional derivative operator with a generalized exponential kernel

On a fractional derivative operator with a singular kernel: definition, properties and numerical simulation

This paper is concerned with proposing a novel nonlocal fractional derivative operator with a singular kernel. We considered a fractional integral operator as a single integral of convolution type combined with a Mittag-Leffler kernel of Prabhakar type. The proposed singular fractional derivative operator is formulated as a proper inverse of the considered integral operator. We provided some useful features and relationships of the proposed derivative and introduced comparisons with the Caputo derivative which can be utilized for potential applications. Next, we presented numerical solutions for some nonlinear fractional order models incorporating the proposed derivative using a numerical algorithm developed in this paper. As a case study, we discussed the dynamic behavior of a fractional logistic model with the proposed derivative.

On the qualitative behavior and numerical analysis of a calcium oscillation model through Atangana–Baleanu operator in Caputo sense

In this research, we regard a miniature significant for many cell types and activates stimulants against external interventions, thereby causing calcium oscillation. We stage the dependent variables regarding the Atangana–Balenau derivative operator in terms of Caputo so that the model can be physically interpreted better. The study investigates the existence and uniqueness of solutions using fixed point theory, stability in terms of Hyers–Ulam, positivity and numerical solutions. In addition, spacetime, phase diagrams and three-dimensional graphic simulations are presented due to the Lagrange interpolation technique for parameter selections that match the existence and uniqueness condition of the solution.

CERTAIN BILINEAR GENERATING RELATIONS FOR THE EXTENDED GAUSS HYPERGEOMETRIC FUNCTION, USING FRACTIONAL CALCULUS

The main aim of the present paper is to apply the extended Riemann Liouville fractional derivative operator for finding some bilinear generating relations for extended Gauss hypergeometric function. Two main results are obtained, which are presented in the form of two theorems.

Some 𝑞-identities derived by the ordinary derivative operator

In this paper, we investigate applications of the ordinary derivative operator, instead of the q q -derivative operator, to the theory of q q -series. As main results, many new summation and transformation formulas are established which are closely related to some well-known formulas such as the q q -binomial theorem, Ramanujan’s 1 ψ 1 {}_1\psi _1 formula, the quintuple product identity, Gasper’s q q -Clausen product formula, and Rogers’ 6 ϕ 5 {}_6\phi _5 formula, etc. Among these results is a finite form of the Rogers-Ramanujan identity and a short way to Eisenstein’s theorem on Lambert series.

Modeling Klebsiella pneumonia infections and antibiotic resistance dynamics with fractional differential equations: insights from real data in North Cyprus

This study presents an enhanced fractional-order mathematical model for analyzing the dynamics of Klebsiella pneumonia infections and antibiotic resistance over time. The model incorporates fractional Caputo derivative operators and kernel, to provide a more comprehensive understanding of the complex temporal dynamics. The model consists of three groups: Susceptible (S), Infected (I), and Resistant (R) individuals, each controlled by a fractional differential equation. The model represents the interaction between infection, recovery from infection, and the possible development of antibiotic resistance in susceptible individuals. The existence, uniqueness, stability, and alignment of the model’s prediction to the observed data were analyzed and buttressed with numerical simulations. The results show that imipenem has the highest efficacy compared with ertapenem and meropenem category drugs. The estimated reproduction number and reproduction coefficient illustrate the potential impact of this model in improving treatment strategies, while the memory effects highlight the advantages of fractional differentiation. The model predicts an increased possibility of antibiotic resistance despite effective treatment, suggesting a new treatment approach.

Analysis of the nonlinear Fitzhugh–Nagumo equation and its derivative based on the Rabotnov fractional exponential function

The Fitzhugh–Nagumo equation is an essential governing equation that explains the process of impulse transmission from the nerves. This work suggests a novel generalized arbitrary-order Fitzhugh–Nagumo equation that is based on the recently developed nonsingular fractional operator. The modified Fitzhugh–Nagumo equation has been generalized using fractional Yang–Abdel–Cattani operator to provide a more accurate representation of the equation with memory effects and non-local behavior. Further, we discussed some interesting and new properties of the proposed fractional operator and the Rabotnov exponential fractional kernel with a modified Sumudu integral transform. In order to solve the proposed nonlinear differential equation, the homotopy perturbation method coupled with the modified Sumudu integral transform technique is implemented to examine the result of a newly proposed Yang–Abdel–Cattani fractional Fitzhugh–Nagumo equation, which converges to the exact solution. The nonlinear terms of this equation are handled in terms of He’s polynomials. The existence and uniqueness of the solution have been demonstrated using the Banach space fixed point theorem. The convergence and error analysis of the suggested technique are also discussed. The findings are expressed in terms of generalized Mittag-Leffler functions. The obtained results are plotted with the help of MATLAB R2016a mathematical software. The results in this work are more accurate, and it is proposed that the new Yang–Abdel–Cattani fractional derivative operator is an effective tool for finding the results of any other nonlinear problems arising in science and engineering. All three problems have been computed for different orders to confirm the significance of the fractionalizing concept in the proposed equation. The increase of the fractional order value ℘ to 1 confirms that our proposed time-fractional Fitzhugh–Nagumo equation in the Yang–Abdel–Cattani sense gets close to the analytic solution.

Second-order Hankel determinant for a subclass of analytic functions satisfying subordination condition connected with modified q-Opoola derivative operator

This paper introduces a new subclass of analytic functions employing the operator that was recently defined by the authors. The coefficients estimate $|a_s| (s = 2, 3)$ of the Taylor-Maclaurin series in this new class, as well as the Fekete-Szegö functional problems, have been derived. Furthermore, we obtained the sharp upper bound for the functional $|a_2a_4 − a_{3}^2|$ for functions belonging to this new subclass

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