Polynomial Solutions Research Articles

SOME NEW REMARKS ON POWER SUMS

Power sum is one of the interesting topics in number theory where its application in other sciences is known to be wide. This paper intends to stipulate new remarks on an explicit polynomial solution to power sums. Additionally, it investigates the general solution under odd and even numbers of terms and discusses some examples.

Trigonometric Polynomial Solutions of Bernoulli Trigonometric Polynomial Differential Equations

We consider real trigonometric polynomial Bernoulli equations of the form A(θ)y′=B1(θ)+Bn(θ)yn where n≥2, with A,B1,Bn being trigonometric polynomials of degree at most μ≥1 in variables θ and Bn(θ)≢0. We also consider trigonometric polynomials of the form A(θ)yn−1y′=B0(θ)+Bn(θ)yn where n≥2, being A,B0,Bn trigonometric polynomials of degree at most μ≥1 in the variable θ and Bn(θ)≢0. For the first equation, we show that when n≥4, it has at most 3 real trigonometric polynomial solutions when n is even and 5 real trigonometric polynomial solutions when n is odd. For the second equation, we show that when n≥3, it has at most 3 real trigonometric polynomial solutions when n is odd and 5 real trigonometric polynomial solutions when n is even. We also provide trigonometric polynomial equations of the two types mentioned above where the maximum number of trigonometric polynomial solutions is achieved. The proof method will be to apply extended Fermat problems to polynomial equations.

The novel cubic B-spline method for fractional Painlevé and Bagley-Trovik equations in the Caputo, Caputo-Fabrizio, and conformable fractional sense

In this analysis, we use the high order cubic B-spline method to create approximating polynomial solutions for fractional Painlevé and Bagley-Torvik equations in the Captuo, Caputo-Fabrizio, and conformable fractional sense concerning boundary set conditions. Using a piecewise spline of a 3rd-degree polynomial; the discretization of the utilized fractional model problems is gained. Taking advantage of the Taylor series expansion; the error order behavior spline theorem is proved. We demonstrate applications of our spline method to several certain kinds including the 1st(2nd) Painlevé and Bagley-Torvik fractional models. For more detail, using Mathematica 11 several drawings and many tables were calculated and their explanations were mentioned. The computational results indicate that the suggested spline approach is most acceptable in terms of cost efficiency and precision of calculations. Highlight, conclusion, and future notes are provided to extract the ability of the discussed approach and the tendency of the utilized fractional models to extrapolate new application areas in the meshless numerical training.

A new perspective to discuss Korteweg-de Vries-like equation

With the help of the generalized bilinear derivative accompanied by the prime number p=3, the formulation of a new Korteweg-de Vries-like equation, was derived. Then, we obtained three classes of unequal rational solutions through consider the rational polynomial solutions to its generalized bilinear form. Lastly, we discussed the rational solutions of this derived equation by 3D and 2D plots.

Mathematical Assessment of Convection and Diffusion Analysis for A Non-Circular Duct Flow with Viscous Dissipation: Application of Physiology

The present analysis has interesting applications in physiology, industry, engineering and medicine. Peristaltic pumps acquire an elliptical cross-section during motion. Peristaltic pumps, roller pumps and finger pumps also have highly useful applications. Transportation through these pumps provides an effective fluid movement and the substance remains separate from the duct walls. Convection and diffusion analyses were executed with accentuated viscous dissipation for the non-Newtonian flow that occurs inside a duct. The viscous effects are reviewed with an integrated convection and diffusion analysis that elucidates in-depth heat flux. Viscous dissipation appears to be the primary cause of increased heat generation. The Cartesian coordinate system is availed to develop this problem under consideration. A dimensionless set of coupled partial differential equations is attained by utilizing the relevant transformations that eventually simplify this complex problem. These coupled equations are solved step by step with a consideration of a polynomial solution method for coupled equations. The unfolded graphical outcomes of velocity, temperature and concentration reveal an axial symmetric flow. A higher rate of convection is observed due to viscous effects. Both the velocity and temperature profiles have an increasing function of Q.

Analytical Solutions Formulated in the Time Domain for Three-Dimensional Heat Diffusion Equation

Two different strategies are provided to generate solutions to the three-dimensional heat diffusion equation. The first strategy is inspired by the well-known one-dimensional heat polynomial, which consists of an infinite set of polynomials, which are solutions to the one-dimensional heat diffusion equation. The second strategy is based on an exponential type function. None of the solutions presented here can be obtained by the method of separation of variables. The mathematical developments proving that, indeed, the particular solutions generated with both strategies satisfy the three-dimensional heat diffusion equation are presented. The analytical solutions are validated by generating the corresponding numerical solutions with the method of finite differences. When comparing both analytical and numerical solutions, it is found that they are identical. In addition, as part of the results, it is found that there are exponential solutions that reproduce the behavior of polynomial solutions. Finally, an example of the use of heat polynomials in engineering applications is provided.

High-Performance Polynomial Root Finding for Graphics

We present a computationally-efficient and numerically-robust algorithm for finding real roots of polynomials. It begins with determining the intervals where the given polynomial is monotonic. Then, it performs a robust variant of Newton iterations to find the real root within each interval, providing fast and guaranteed convergence and satisfying the given error bound, as permitted by the numerical precision used. For cubic polynomials, the algorithm is more accurate and faster than both the analytical solution and directly applying Newton iterations. It trivially extends to polynomials with arbitrary degrees, but it is limited to finding the real roots only and has quadratic worst-case complexity in terms of the polynomial's degree. We show that our method outperforms alternative polynomial solutions we tested up to degree 20. We also present an example rendering application with a known efficient numerical solution and show that our method provides faster, more accurate, and more robust solutions by solving polynomials of degree 10.

A highly accurate matrix method for solving a class of strongly nonlinear BVP arising in modeling of human shape corneal

This research study presents a novel highly accurate matrix approach for numerical treatments of a strongly nonlinear boundary value problem occurring in the modeling of the corneal shape of the human eye. Using the technique of quasilinearization, the nonlinear model is reduced into a sequence of linearized problems. Then, a spectral collocation procedure based on novel shifted Vieta‐Fibonacci (SVF) is applied to transform each subproblem into a linear algebraic system of equations. An upper bound for the error is provided, and convergence analysis of the SVF series solution in the weighted norm is discussed. The technique of error correction is used to improve the SVF polynomial solutions by means of the residual error function. Various numerical tests are provided to demonstrate the accuracy and efficiency of the presented collocation algorithm. The validation of the proposed approach is shown by comparison with available existing numerical solutions.

Rational solutions of Abel trigonometric polynomial differential equations

In this paper we deal with the trigonometric polynomial differential equations of the form Y′=A(θ)Y2+B(θ)Y3, where A and B are real trigonometric polynomials with B(θ)≢0. We first prove that these equations has at most two trigonometric polynomial solutions (which will always be constant solutions) and show that this upper bound is reached. Second we provide an upper bound on the number of rational trigonometric solutions that these equations can have and we show that under some conditions related with the degree of the trigonometric polynomials A(θ) and B(θ), the number of rational solutions is two.

CONSTRUCTION OF A MULTIVARIATE POLYNOMIAL GIVEN BY A REDUNDANT DESCRIPTION IN STOCHASTIC AND DETERMINISTIC FORMULATIONS USING AN ACTIVE EXPERIMENT

We present the methods for constructing a multivariate polynomial given by a redundant representation based on the results of a limited active experiment. We solve the problem in two formulations. The first is the problem of constructing a multivariate polynomial regression given by a redundant representation based on the results of a limited active experiment. The solution method is based on the previous results of Professor A. A. Pavlov and his students showing the fundamental possibility of reducing this problem to the sequential construction of univariate polynomial regressions and solving the corresponding nondegenerate systems of linear equations. There are two modifications of this method. The second modification is based on proving for an arbitrary limited active experiment the possibility of using only one set of normalized orthogonal polynomials of Forsythe. The second formulation refers to the solution of this problem for a particular but sufficient from the practical point of view case when an unknown implementation of a random variable is not added to the initial measurement results during an active experiment. This method is a modification of the solution method for the multivariate polynomial regression problem. Also, we used the main results of the general theory (which reduces the multivariate polynomial regression problem solving to the sequential construction of univariate polynomial regressions and solution of corresponding nondegenerate systems of linear equations) to consider and strictly substantiate fairly wide from the practical point of view particular cases leading to estimating the coefficients at nonlinear terms of the multivariate polynomial regression as a solution of linear equations with a single variable.

ANOTHER LOOK AT A POLYNOMIAL SOLUTION FOR ALTERNATING POWER SUMS

This paper aims to provide a new general explicit polynomial solution on power sums of consecutive positive integers under alternating signs. Moreover, it examines the solution under odd and even number of terms in the series and provides some examples.

Optical solitons and qualitative analysis of nonlinear Schrodinger equation in the presence of self steepening and self frequency shift

In this manuscript, the optical solitons of nonlinear Schrödinger equation (NLSE) with cubic–quintic law nonlinearity, in the presence of self-frequency shift and self-steepening, has been studied. The ultrahigh capacity propagation and transit of laser light pulses in optical fibres were described using this form of equation. To extract new results, two strong methodologies has been used. To extract the exact solution unified method has been employed. The solutions obtained by this analytical method, are in form of polynomial and rational function solution. Moreover, the validity of non-singular solutions has guaranteed by a limitation condition that is graphically illustrated in 3D. The 2D graphical representation are also used to demonstrate the influence of parameters on the predicted non-singular solutions. The other technique, used for qualitative analysis, is bifurcation. The system has been transformed into a planer dynamical system, which has been transformed into a hamiltonian system. All the possible phase portrait has been plotted by complete discrimination method. The acquired results are novel and have not been recorded before and they indicate that the proposed methodologies may be used to investigate innovative soliton solutions and phase portraits for any NLSE.

Sunset similarity solution for a receding hydraulic fracture

This paper derives approximate ‘sunset’ similarity solutions for receding plane strain and radially symmetric hydraulic fractures in permeable elastic media close to the point of closure. Local analysis is used to show that a receding hydraulic fracture has a linear aperture asymptote$\hat {w}\sim \hat {s}$in the fracture tip, where$\hat {s}$is the distance from the fracture front. Due to the regularity of the linear asymptote, it is possible to determine similarity solutions in the form of power series expansions, which, for integers$N\ge 2$and values of the radius decay exponent$\gamma =1/N$, can be shown to terminate to yield polynomial solutions for the fracture aperture of degree$N$. Of this countable infinity of polynomial solutions, the final aperture profile as the fracture approaches closure is associated with the second-degree polynomial with$\gamma =1/2$called the sunset solution. For the reverse time$t^{\prime }$measured from closure, the sunset solution is characterized by$w\sim t^{\prime }$and$R\sim t^{\prime 1/2}$. Of all the admissible polynomial similarity solutions, the sunset solution is shown to form an attractor, as$t^{\prime }\rightarrow 0$, for receding hydraulic fractures associated with a wide variety of points in parametric space. Using the sunset solution, it is possible to estimate the duration of recession, assuming that the fracture aperture and radius at the start of recession are given, and determine how it scales with a dimensionless shut-in parameter. As the fracture approaches closure, the term responsible for coupling the elastic force balance and fluid conservation becomes subdominant to the other terms in the lubrication equation, which reduces to a local kinematic relation between the decaying fracture aperture and the leak-off velocity. This fundamental decoupling of dynamics from kinematics results in the sunset solution being dependent on only a single material parameter – namely the leak-off coefficient. This isolation of the leak-off coefficient by the sunset solution opens the possibility to determine this parameter from laboratory or field measurements.

Phase characterization and new optical solitons of a pulse passing through nonlinear dispersive media

In this paper, modified equal width Burgers’ equation has been investigated with the aid of unified method and bifurcation. This model has many applications in long wave transmission with dispersion and dissipation in nonlinear medium. The applied technique is efficient to retrieve exact solutions and their dynamic behaviors. The obtained solutions are polynomial and rational function solutions. The behavior of dynamical planer system has been analyzed by assigning different values to the parameters, also each possible case has been shown as phase portraits in this research paper. The estimated solutions demonstrate that the proposed approaches are simple, practical, and promising for investigating further equal width equation’s soliton wave solutions and phase portraits.

New classes of particular solutions to one problem on gyrostat motion

The paper studies the existence of two new classes of polynomial solutions to differential equations related to the problem of the gyrostat motion with a fixed point in the magnetic field, taking into account the Barnett-London effect. A common feature of the structure of these classes is that the functions that set the invariance relations for the unit vector components of the symmetry axis of the active force fields are either rational functions of the first component of the specified vector or of the auxiliary variable. Three new particular solutions to the polynomial classes under consideration are constructed. These solutions are described by the functions obtained by the inversion of hyperelliptic integrals. It has been proved that another constructed solution of the polynomial structures under study, for which the movement of the gyrostat has the property of precession, is a particular case of a known solution.

Single-traveling-wave and double-wave polynomial solutions of the generalized (3 + 1)-dimensional modified Kadomtsev–Petviashvili equations with variable coefficients

In this paper, we explore the generalized (3 + 1)-dimensional modified Kadomtsev–Petviashvili equations with variable coefficients, which are usually used in the fields of ferromagnetism, magneto-optics, plasma physics and fluid mechanics. For the sake of uncovering more physical phenomena related to this system, we consider the single-traveling-wave polynomial solutions in the light of the unified method and the double-wave polynomial solutions in line with the generalized unified method. Remarkably, the solitary-, soliton- as well as elliptic-type solutions are all discussed in these two kinds of solutions. Furthermore, the physical explanations of the solutions are given graphically and analytically for different choices of the free parameters (especially of the nonlinear coefficients of the equations that related to the physical insights). By discussing the wave propagation of each solution that we procured from the perspectives of amplitude, shape, symmetry or periodicity, we are capable of realizing the inherent characteristics of this equation commendably and discovering the correlative physical world more efficiently.

Characterization of rational solutions of a KdV-like equation

We characterize the rational solutions to a KdV-like equation which are generated from polynomial solutions to the corresponding generalized bilinear equation. We use a particular class of polynomials satisfying a quadratic difference equation to obtain that there are no solutions of the bilinear equation with degree (in the spatial variable) greater than 5. As a byproduct, we answer positively a conjecture of Yi Zhang and Wen-Xiu Ma about these solutions.

New Solitons and Multishock Wave Structures for the Conformable Space Fractional Burger and Time Fractional Sharma-Tasso-Olver Models

This present paper studies the conformable space fractional Burgers, and the time fractional Sharma-Tasso-Olver models; both are highly important for nonlinear diffusive waves in fluid dynamics, sound waves in a viscous medium, and flow in field soils, as well as in gas and plasma dynamics. To retrieve explicit solutions of the fractional differential models, we propose an integral scheme, namely, Modified Kudryashov method. We obtain periodic, solitary, mixed periodic-soliton, and polynomial solutions through the approach. In particular, we exhibit topological kink-dark bell wave, topological kink, singular kink, bright bell, peakon solitons, and periodic shape waves to apply suitable values on parameters for both distinct models. The impact of fractionality on the wave shape and its deformation is analyzed and discussed graphically. We also investigate multishock wave’s solutions of both models and analyzed the effect of each existing parameters involved in the obtained solutions. To visualize the real characters of the solitary solutions, the graphical elucidation in 3D and 2D profiles are plotted. In computational effort and realization, it is emphasized that the proposed scheme is friendly useful, highly effective, and a powerful mathematical tool to extract exact solitary wave solutions for the differential models, as well as fractional differential models.

The Orthogonal Branching Problem for Symplectic Monogenics

In this paper we study the $$\mathfrak {sp}(2m)$$ -invariant Dirac operator $$D_s$$ which acts on symplectic spinors, from an orthogonal point of view. By this we mean that we will focus on the subalgebra $$\mathfrak {so}(m) \subset \mathfrak {sp}(2m)$$ , as this will allow us to derive branching rules for the space of 1-homogeneous polynomial solutions for the operator $$D_s$$ (hence generalising the classical Fischer decomposition in harmonic analysis for a vector variable in $${\mathbb {R}}^m$$ ). To arrive at this result we use techniques from representation theory, including the transvector algebra $${\mathcal {Z}}(\mathfrak {sp}(4),\mathfrak {so}(4))$$ and tensor products of Verma modules.

Numerical solutions of the Bagley–Torvik equation by using generalized functions with fractional powers of Laguerre polynomials

Abstract In this study, a collocation approach is presented to solve Bagley–Torvik equation, which is a class of fractional differential equations. As most fractional differential equations do not have exact analytical solutions, it is needed numerical methods. This study is important because it presents a numerical method for fractional differential equations. The main purpose of this method is to obtain the approximate solution based on Laguerre polynomials of the Bagley–Torvik equation. To date, a collocation method based on the Laguerre polynomials has not been studied for the solutions of the Bagley–Torvik equation. This reveals the novelty of the study. The approximate solution is sought in form of the fractional powers of the Laguerre polynomials. By using the Caputo derivative, the matrix relation is created for term with fractional derivative in the equation. Similarly, the matrix relation of second derivative is computed in equation. Then, by using these matrix relations and the collocation points, the Bagley–Torvik problem is converted into a system of the linear algebraic equations. The solution of this system gives the coefficients of the assumed solution. Secondly, an error estimation method is given with the help of the residual function and also the Laguerre polynomial solution is improved by using the estimated error function. Then, the method is applied to four examples and the obtained numerical results are shown in tables and graphs. Also, the comparisons are made with other methods in the literature and so the presented method gives better results than other methods.