Polynomial Type Research Articles

Lump and soliton on certain spatially-varying backgrounds for an integrable (3+1) dimensional fifth-order nonlinear oceanic wave model

The dynamics of nonlinear waves with controllable dispersion, nonlinearities, and background continues to be an exciting line of active research in recent years. In this work, we focus to investigate an integrable (3+1)-dimensional nonlinear model describing the evolution of water waves with higher-order temporal dispersion by characterizing the dynamics of lump and soliton waves on different spatially-varying backgrounds. Particularly, we construct some explicit lump wave and kink-soliton solutions for the considered model through its trilinear and bilinear equations with appropriate forms of the polynomial and exponential type initial seed solutions, respectively. Additionally, we obtain hyperbolic and periodic waves through bilinear Bäcklund transformation. We explore their propagation and transformation/modulation dynamics due to different spatial backgrounds through categorical analysis and clear graphical demonstrations for a complete understanding of the resulting solutions. Our analysis shows that the lump solution results into a simultaneous existence of well-localized spike and declining (coupled bright-dark wave) structures on a constant background, while the soliton solution exhibits kink and anti-kink wave patterns along different spatio-temporal domains with controllable properties. When the arbitrary spatial background is incorporated appropriately, we can witness the manifestation of soliton into manipulated periodic and interacting dynamical structures. On the other hand, the lump wave results into the coexistence, interacting wave and breather formation due to periodic and localized type arbitrary spatial backgrounds. The present results will be an important addition to the context of engineering nonlinear waves with non-vanishing controllable backgrounds in higher-dimensional models. Further, the present study can be extended to investigate several other nonlinear systems to understand the physical insights of the variable background in their dynamics.

Generalized Ismail-Durrmeyer type operators involving Sheffer polynomials

In this paper, we introduce a new kind of Durrmeyer variant of generalized Ismail operators based on Sheffer polynomials. These type of polynomials are a generalization of Appell polynomials and a special case of Boas-Buck-type polynomials. We study the convergence of these operators with the help of universal Korovkin type theorem, modulus of continuity, Peetre's $ K $-functional and the class of Lipschitz type functions. Furthermore, We extend this study to include Voronovskaja-type asymptotic results, quantitative-Voronovskaja and Gr$ \ddot{u} $ss-Voronovskaja type theorems and the convergence rate of these operators for functions in a polynomial weighted space by means of the weighted modulus of continuity.

Delayed wave equation with logarithmic variable-exponent nonlinearity

<abstract><p>A delayed nonlinear wave equation with variable exponents of logarithmic type is discussed in this paper. In the presence of the logarithmic nonlinear source, we established a global existence result under sufficient conditions on the initial data only without imposing the Sobolev Logarithmic Inequality. After that, we established global results of exponential and polynomial types according to the range values of the exponents. At the end, we give a numerical study that supports our theoretical results.</p></abstract>

Some identities of degenerate higher-order Daehee polynomials based on $ \lambda $-umbral calculus

<abstract><p>The degenerate versions of special polynomials and numbers, initiated by Carlitz, have regained the attention of some mathematicians by replacing the usual exponential function in the generating function of special polynomials with the degenerate exponential function. To study the relations between degenerate special polynomials, $ \lambda $-umbral calculus, an analogue of umbral calculus, is intensively applied to obtain related formulas for expressing one $ \lambda $-Sheffer polynomial in terms of other $ \lambda $-Sheffer polynomials. In this paper, we study the connection between degenerate higher-order Daehee polynomials and other degenerate type of special polynomials. We present explicit formulas for representations of the polynomials using $ \lambda $-umbral calculus and confirm the presented formulas between the degenerate higher-order Daehee polynomials and the degenerate Bernoulli polynomials, for example. Additionally, we investigate the pattern of the root distribution of these polynomials.</p></abstract>

Alternation points, weights and orthogonal polynomials

In this note we characterize the orthogonal polynomials having alternation points in terms of algebraic properties of the coefficients of their recurrence relations. As an example, we deduce that the four types of Chebyshev polynomials are the only Jacobi polynomials of degree greater than 3 that have alternation points. We also show that if there are alternation points for polynomials orthogonal with respect to a measure μ , then the inner product defined by μ is a multiple of the discrete inner product generated by the alternation points. Finally, we produce orthonormal polynomials having alternation points that are the roots of a given trigonometric equation.

RESULTS ON UNIQUENESS OF L-FUNCTIONS CONCERNING WEIGHTED SHARING

In this paper using the notion of weighted sharing, we consider the value distribution of a L- function and meromorphic function when certain type of difference-differential polynomials which share a small function and rational function and obtain some uniqueness results which extends recent results due to Hao and Chen[2].

On stability and regularity for semilinear anomalous diffusion equations perturbed by weak-valued nonlinearities

Various diffusion processes in media with memory are depicted by anomalous diffusion equations. We are concerned with a class of anomalous diffusion equations with the nonlinearity taking values in Hilbert scales of negative order. The fundamental questions on global solvability, stability and regularity of solutions to the Cauchy problem governed by the mentioned equations are taken into account. We obtain some solvability results under different assumptions on the regularity of the nonlinear function. When the nonlinearity becomes less singular, the asymptotic stability of solutions is proved. Provided that the kernel function is 2-regular and sectorial, we show the Hölder continuity of the obtained solutions. Our approach is based on the resolvent theory, fixed point argument and embeddings of fractional Sobolev spaces. The applicability of our results is presented for some anomalous diffusion problems, where the nonlinear function is of polynomial or convection type.

On the parabolic Cauchy problem for quantum graphs with vertex noise

We investigate the parabolic Cauchy problem associated with quantum graphs including Lipschitz or polynomial type nonlinearities and additive Gaussian noise perturbed vertex conditions. The vertex conditions are the standard continuity and Kirchhoff assumptions in each vertex. In the case when only Kirchhoff conditions are perturbed, we can prove existence and uniqueness of a mild solution with continuous paths in the standard state space $\mathcal{H}$ of square integrable functions on the edges. We also show that the solution is Markov and Feller. Furthermore, assuming that the vertex values of the normalized eigenfunctions of the self-adjoint operator governing the problem are uniformly bounded, we show that the mild solution has continuous paths in the fractional domain space associated with the Hamiltonian operator, $\mathcal{H}_{\alpha}$ for $\alpha<\frac{1}{4}$. This is the case when the Hamiltonian operator is the standard Laplacian perturbed by a potential. We also show that if noise is present in both type of vertex conditions, then the problem admits a mild solution with continuous paths in the fractional domain space $\mathcal{H}_{\alpha}$ with $\alpha<-\frac{1}{4}$ only. These regularity results are the quantum graph analogues obtained by da Prato and Zabczyk [9] in case of a single interval and classical boundary Dirichlet or Neumann noise.

Mittag-Leffler stability of numerical solutions to linear homogeneous time fractional parabolic equations

&lt;abstract&gt;&lt;p&gt;In a bounded domain, the solution of linear homogeneous time fractional parabolic equation is known to exhibit polynomial type decay rate (the so-called Mittag-Leffler stability) over time, which is quite different from the exponential decay of classical parabolic equation. We firstly use the finite element method or finite difference method to discretize the parabolic equation in space to obtain fractional ordinary differential equation, and then use fractional linear multistep method (F-LMM) to discretize in time to obtain a fully discretized schemes. We prove that the strongly $ A $-stable F-LMM method combined with appropriate spatial discretization can accurately maintain the long-term optimal algebraic decay rate of the original continuous equation. Numerical examples are included to confirm the correctness of our theoretical analysis.&lt;/p&gt;&lt;/abstract&gt;

Aluminum Deoxidation Equilibrium in Molten Fe-Co Alloys

Aluminum deoxidation equilibrium of molten Fe-Co alloy was experimentally measured using a chemical equilibrium method and numerically assessed using a sub-regular solution model based on Darken’s quadratic formalism and a Redlich-Kister type polynomial at 1873 K. It was found that the degree of oxygen content reduction by Al-deoxidation decreased with increasing cobalt concentration in the alloy, peaking at around 40 to 60 mass% Co, and then improved with further increase in cobalt concentration. The following binary interaction parameters between cobalt and aluminum were derived in this study:

Electrostatic Partners and Zeros of Orthogonal and Multiple Orthogonal Polynomials

For a given polynomial P with simple zeros, and a given semiclassical weight w, we present a construction that yields a linear second-order differential equation (ODE), and in consequence, an electrostatic model for zeros of P. The coefficients of this ODE are written in terms of a dual polynomial that we call the electrostatic partner of P. This construction is absolutely general and can be carried out for any polynomial with simple zeros and any semiclassical weight on the complex plane. An additional assumption of quasi-orthogonality of P with respect to w allows us to give more precise bounds on the degree of the electrostatic partner. In the case of orthogonal and quasi-orthogonal polynomials, we recover some of the known results and generalize others. Additionally, for the Hermite–Padé or multiple orthogonal polynomials of type II, this approach yields a system of linear second-order differential equations, from which we derive an electrostatic interpretation of their zeros in terms of a vector equilibrium. More detailed results are obtained in the special cases of Angelesco, Nikishin, and generalized Nikishin systems. We also discuss the discrete-to-continuous transition of these models in the asymptotic regime, as the number of zeros tends to infinity, into the known vector equilibrium problems. Finally, we discuss how the system of obtained second-order ODEs yields a third-order differential equation for these polynomials, well described in the literature. We finish the paper by presenting several illustrative examples.

ON THE DYNAMICS OF THE HODGKIN-HUXLEY NEURAL MODEL: A STABLE ANALYSIS USING HURWITZ POLYNOMIALS

In this paper, a study of the dynamics of the Hodgkin-Huxley neural model is conducted. This stability study is performed using the stability criteria for obtaining Hurwitz polynomials that provide necessary and/or sufficient conditions to analyze the dynamics of the model by studying the location of the roots of the characteristic polynomial associated with it. The main objective of the article is to analyze the stability of the Hodgkin-Huxley neural model using an analytical technique to establish Hurwitz type polynomials. This article describes and presents an analytical method to analyze the stability of the Hodgkin-Huxley neural model based on obtaining Hurwitz polynomials through stability criteria. This technique allows establishing the asymptotic stability of the neural model through the localization of the Eigen values of the matrix associated with the model. The results of this research allow establishing asymptotic stability conditions of the Hodgkin-Huxley neural model that allow analyzing the dynamics of the model. Through this article, it is possible to analyze the dynamics of the Hodgkin-Huxley neural model and allows drawing inferences regarding the initiation and transmission of action potentials in neurons and cardiac cells using the Hurwitz polynomial technique.

The algebra of difference operators associated to Meixner type polynomials

Meixner type polynomials are defined from the Meixner polynomials by using Casoratian determinants whose entries belong to two given finite sets of polynomials and . They are eigenfunctions of higher-order difference operators but only for a careful choice of the polynomials and , the sequence is orthogonal with respect to a measure. Under mild assumptions, we characterize in this paper the algebra formed by all difference operators with respect to which the family of Meixner type polynomials are eigenfunctions.

A new class of Laguerre based Frobenius type Eulerian numbers and polynomials

In this article, we introduce a new class of generalized Laguerre-based Frobenius type Eulerian polynomials and then derive diverse explicit and implicit summation formulae and symmetric identities by using series manipulation techniques. Multifarious summation formulas and identities are given earlier for some well known polynomials such as Eulerian polynomials and Frobenius type Eulerian polynomials are generalized.

Location of zeros of a Lacunary type polynomial

For a given polynomial p(z) of degree n with real or complex coefficients, our basic aim has been to determine the smallest region in which all the zeros of p(z) lie. In the present paper, we have obtained a result by using Lacunary type polynomial which gives the region of zeros neither circular nor annular except in some particular cases. Our result plays an important role to reduce the region of polynomial zeros.

On the Hosoya polynomial of the third type of the chain hex-derived network

A topological index plays an important role in characterising various physical properties, biological activities, and chemical reactivities of a molecular graph. The Hosoya polynomial is used to evaluate the distance-based topological indices such as the Wiener index, hyper-Wiener index, Harary index, and Tratch – Stankevitch – Zefirov index. In the present study, we determine a closed form of the Hosoya polynomial for the third type of the chain hex-derived network of dimension n and derive the distance-based topological indices of the network with the help of their direct formulas and alternatively via using the obtained Hosoya polynomial. Finally, we graphically represent the computed distance-based topological indices and the Hosoya polynomial of the underlying network to comprehend their geometrical pattern. This study of the Hosoya polynomial and the corresponding indices can set the basis for more exploration into chain hex-derived networks and their properties.

A derivative-type operator and its application to the solvability of a nonlinear three point boundary value problem

In this paper we introduce an operator that can be thought as a derivative of variable order, i.e. the order of the derivative is a function. We prove several properties of this operator, for instance, we obtain a generalized Leibniz’s formula, Rolle and Cauchy’s mean theorems and a Taylor type polynomial. Moreover, we obtain its inverse operator. Also, with this derivative we analyze the existence of solutions of a nonlinear three-point boundary value problem of “variable order”.

Updated Spectral Type Polynomials for Ultracool Dwarfs with CatWISE Photometry

Using a volume-limited sample of 263 field-age brown dwarfs defined by parallaxes, we provide updated polynomials for CatWISE W1 and W2 absolute magnitudes as a function of spectral type. Our fits are consistent with previous relations and apply to a wider M6–Y1 spectral type range.

Invariant soliton solutions for the coupled nonlinear Schrödinger type equation

The Schrödinger equation is an essential model in quantum mechanics. It simulates fascinating nonlinear physical phenomena, such as shallow-water waves, hydrodynamics, harmonic oscillator, nonlinear optics, and quantum condensates. The purpose of this study is to look at the optical soliton solutions to nonlinear triple-component Schrödinger equations using the Lie classical approach combined with modified (G′/G)-expansion method and polynomial type assumption. As a result of these approaches, some explicit solutions such as hyperbolic, periodic, and power series solutions are found. In addition, we look at the stability of the corresponding to one of the reductions using phase plane theory. Maple software is used to graphically represent some of the acquired solitons and phase portraits. Compared to the other techniques, we can conclude that the current methods are effective, powerful, and provide simple, trustworthy solutions. Maple software was used to check all of the obtained solutions.

Thermodynamic Properties of Binary Liquid Mixtures Containing Furfural with Chlorobenzene, Nitromethane, Diethylmalonate and 1-Butanol at 308.15K and 318.15K Supported by FTIR Spectral Studies

A study on thermodynamic properties of the liquid mixture is used in the industrial process, which often extends to solution chemistry. Density(ρ), Viscosity(η), and Ultrasonic velocity(U) were determined for binary liquid mixtures of furfural with chlorobenzene, nitromethane, diethyl malonate and 1- butanol at temperature of 308.15K and 318.15K at atmospheric pressure over the whole range of mole fractions. The calculated thermodynamic properties and some excess parameters such as Excess Volume(VE), Deviation in Isentropic Compressibility(∆KS), Deviation in Viscosity(∆η), Deviation in Intermolecular Free Length(∆LF), Deviation in Intermolecular Free Volume(∆VF), Deviation in Internal Pressure(∆π) and Deviation in Acoustic Impedance(∆Z) were calculated and applied to the Redlich-Kister type polynomial equation to determine the appropriate coefficients. The effects of composition and temperature on thermodynamic parameters have been studied in terms of molecule interaction in these liquid mixtures. Further, IR spectra of these liquid mixtures were recorded and the data were utilized to examine the mixing behavior of the components.