Th-degree Polynomial Research Articles

Study of the Six-Compartment Nonlinear COVID-19 Model with the Homotopy Perturbation Method

The current study aims to utilize the homotopy perturbation method (HPM) to solve nonlinear dynamical models, with a particular focus on models related to predicting and controlling pandemics, such as the SIR model. Specifically, we apply this method to solve a six-compartment model for the novel coronavirus (COVID-19), which includes susceptible, exposed, asymptomatic infected, symptomatic infected, and recovered individuals, and the concentration of COVID-19 in the environment is indicated by S(t), E(t), A(t), I(t), R(t), and B(t), respectively. We present the series solution of this model by varying the controlling parameters and representing them graphically. Additionally, we verify the accuracy of the series solution (up to the (n−1)th-degree polynomial) that satisfies both the initial conditions and the model, with all coefficients correct at 18 decimal places. Furthermore, we have compared our results with the Runge–Kutta fourth-order method. Based on our findings, we conclude that the homotopy perturbation method is a promising approach to solve nonlinear dynamical models, particularly those associated with pandemics. This method provides valuable insight into how the control of various parameters can affect the model. We suggest that future studies can expand on our work by exploring additional models and assessing the applicability of other analytical methods.

A 〖(2n+9)〗^thDegree Polynomial Approximation Method for Solving Eight Order Linear Boundary Value Problems

Galerking method, Collocation method and least squired method have been used for solving boundary value problems (BVPs) and from practical point of view almost any success of above mention method depends on the selection of basis functions. Also, Adomain decomposition and variational iteration methods have been applied for the solution of BVPs and required respectively calculating the Adomian’s polynomials and Lagrange multipliers. In this paper a simple method of Degree polynomial approximation solution were n is number of successive derivatives of the governing equation is suggested as an alternative of the above mentioned methods. The method consists of first obtaining from the governing equation its n successive derivatives and boundary conditions a linear differential system of equations evaluated at the boundary points. Next, an approximate solution in the form of polynomial of degree with unknown coefficients is assumed. To determine the unknown coefficients, the assumed solution is incorporate into the linear differential system which turns to be a linear system of equation with unknowns which is salvable uniquely. The method is tested for n=5 to four examples. It clear that from tables 3.1 to 3.4 and figures 3.1 to 3.4 the numerical outcomes agree with the exact solution and also better than some existing results in the literatures. it should be also noted that the accuracy of the method can be increased by increased n. All Numerical computation were performed using maple 2016 software.

The Stability and Bifurcation of the (2m+1)th-Degree Polynomial Systems

The Stability and Bifurcation of the (2m+1)th-Degree Polynomial Systems

The Stability and Bifurcation of the (2m)th-degree Polynomial Systems

In this paper, the global stability and bifurcations of equilibriums in the (2m)th-degree polynomial system are discussed for a better understanding the complexity of bifurcations and stability of equilibriums. The appearing and switching bifurcations for simple equilibriums are presented, and the appearing and switching bifurcations for higherorder equilibriums are discussed as well. The teethcomb-appearing, spraying-appearing, and sprinkler-spraying-appearing bifurcations for simple and higher-order equilibriums are presented. The antennaswitching bifurcations for simple and higher-order equilibriums are discussed and the parallel straw-bundle-switching and flower-bundleswitching bifurcations for simple and higher-order equilibriums are presented as well.

QUASI-EXACTLY SOLVABLE SCHRÖDINGER EQUATIONS, SYMMETRIC POLYNOMIALS AND FUNCTIONAL BETHE ANSATZ METHOD

For applications to quasi-exactly solvable Schrödinger equations in quantum mechanics, we consider the general conditions that have to be satisfied by the coefficients of a second-order differential equation with at most <em>k </em>+ 1 singular points in order that this equation has particular solutions that are <em>n</em>th-degree polynomials. In a first approach, we show that such conditions involve <em>k </em>- 2 integration constants, which satisfy a system of linear equations whose coefficients can be written in terms of elementary symmetric polynomials in the polynomial solution roots whenver such roots are all real and distinct. In a second approach, we consider the functional Bethe ansatz method in its most general form under the same assumption. Comparing the two approaches, we prove that the above-mentioned <em>k </em>- 2 integration constants can be expressed as linear combinations of monomial symmetric polynomials in the roots, associated with partitions into no more than two parts. We illustrate these results by considering a quasi-exactly solvable extension of the Mathews-Lakshmanan nonlinear oscillator corresponding to <em>k </em>= 4.

The Local Discontinuous Galerkin Method for the Fourth-Order Euler–Bernoulli Partial Differential Equation in One Space Dimension. Part I: Superconvergence Error Analysis

In this paper we develop and analyze a new superconvergent local discontinuous Galerkin (LDG) method for approximating solutions to the fourth-order Euler---Bernoulli beam equation in one space dimension. We prove the $$L^2$$ L 2 stability of the scheme and several optimal $$L^2$$ L 2 error estimates for the solution and for the three auxiliary variables that approximate derivatives of different orders. Our numerical experiments demonstrate optimal rates of convergence. We also prove superconvergence results towards particular projections of the exact solutions. More precisely, we prove that the LDG solution and its spatial derivatives (up to third order) are $$\mathcal O (h^{k+3/2})$$ O ( h k + 3 / 2 ) super close to particular projections of the exact solutions for $$k$$ k th-degree polynomial spaces while computational results show higher $$\mathcal O (h^{k+2})$$ O ( h k + 2 ) convergence rate. Our proofs are valid for arbitrary regular meshes and for $$P^k$$ P k polynomials with $$k\ge 1$$ k ? 1 , and for periodic, Dirichlet, and mixed boundary conditions. These superconvergence results will be used to construct asymptotically exact a posteriori error estimates by solving a local steady problem on each element. This will be reported in Part II of this work, where we will prove that the a posteriori LDG error estimates for the solution and its derivatives converge to the true errors in the $$L^2$$ L 2 -norm under mesh refinement.

On the de la Garza phenomenon

Deriving optimal designs for nonlinear models is, in general, challenging. One crucial step is to determine the number of support points needed. Current tools handle this on a case-by-case basis. Each combination of model, optimality criterion and objective requires its own proof. The celebrated de la Garza Phenomenon states that under a (p−1)th-degree polynomial regression model, any optimal design can be based on at most p design points, the minimum number of support points such that all parameters are estimable. Does this conclusion also hold for nonlinear models? If the answer is yes, it would be relatively easy to derive any optimal design, analytically or numerically. In this paper, a novel approach is developed to address this question. Using this new approach, it can be easily shown that the de la Garza phenomenon exists for many commonly studied nonlinear models, such as the Emax model, exponential model, three- and four-parameter log-linear models, Emax-PK1 model, as well as many classical polynomial regression models. The proposed approach unifies and extends many well-known results in the optimal design literature. It has four advantages over current tools: (i) it can be applied to many forms of nonlinear models; to continuous or discrete data; to data with homogeneous or nonhomogeneous errors; (ii) it can be applied to any design region; (iii) it can be applied to multiple-stage optimal design and (iv) it can be easily implemented.

Solvable Rational Potentials and Exceptional Orthogonal Polynomials in Supersymmetric Quantum Mechanics

New exactly solvable rationally-extended radial oscillator and Scarf I potentials are generated by using a constructive supersymmetric quantum mechanical method based on a reparametrization of the corresponding conventional superpotential and on the addition of an extra rational contribution expressed in terms of some polynomial g. The cases where g is linear or quadratic are considered. In the former, the extended potentials are strictly isospectral to the conventional ones with reparametrized couplings and are shape invariant. In the latter, there appears a variety of extended potentials, some with the same charac- teristics as the previous ones and others with an extra bound state below the conventional potential spectrum. Furthermore, the wavefunctions of the extended potentials are con- structed. In the linear case, they contain ( + 1)th-degree polynomials with = 0,1,2,..., which are shown to beX1-Laguerre orX1-Jacobi exceptional orthogonal polynomials. In the quadratic case, several extensions of these polynomials appear. Among them, two different kinds of ( + 2)th-degree Laguerre-type polynomials and a single one of ( + 2)th-degree Jacobi-type polynomials with = 0,1,2,... are identified. They are candidates for the still unknown X2-Laguerre and X2-Jacobi exceptional orthogonal polynomials, respectively.

Measuring polynomial functions of states

In this paper I show that any $m$th-degree polynomial function of the elements of the density matrix $\rho$ can be determined by finding the expectation value of an observable on $m$ copies of $\rho$, without performing state tomography. Since a circuit exists which can approximate the measurement of any observable, in principle one can find a circuit which will estimate any such polynomial function by averaging over many runs. I construct some simple examples and compare these results to existing procedures.

Expansions for the Fundamental Hermite Interpolation Polynomials in Terms of Chebyshev Polynomials

We obtain explicit expansions of the fundamental Hermite interpolation polynomials in terms of Chebyshev polynomials in the case where the nodes considered are either zeros of the (n + 1)th-degree Chebyshev polynomial or extremum points of the nth-degree Chebyshev polynomial.

Spectrum generating algebra of the Cλ-extended oscillator and multiphoton coherent states

The C λ -extended oscillator spectrum generating algebra is shown to be a C λ -extended ( λ−1)th-degree polynomial deformation of su(1,1). Its coherent states are constructed. Their statistical and squeezing properties are studied in detail. Such states include both some Barut–Girardello and the standard λ-photon coherent states as special cases.

Quasi-exactly solvable quartic potential

A new two-parameter family of quasi-exactly solvable quartic polynomial potentials $V(x)=-x^4+2iax^3+(a^2-2b)x^2+2i(ab-J)x$ is introduced. Until now, it was believed that the lowest-degree one-dimensional quasi-exactly solvable polynomial potential is sextic. This belief is based on the assumption that the Hamiltonian must be Hermitian. However, it has recently been discovered that there are huge classes of non-Hermitian, ${\cal PT}$-symmetric Hamiltonians whose spectra are real, discrete, and bounded below [physics/9712001]. Replacing Hermiticity by the weaker condition of ${\cal PT}$ symmetry allows for new kinds of quasi-exactly solvable theories. The spectra of this family of quartic potentials discussed here are also real, discrete, and bounded below, and the quasi-exact portion of the spectra consists of the lowest $J$ eigenvalues. These eigenvalues are the roots of a $J$th-degree polynomial.

Euler and the Fundamental Theorem of Algebra

A watershed event for all students of mathematics is the first course in basic high school algebra. In my case, this provided an initial look at graphs, inequalities, the quadratic formula, and many other critical ideas. Somewhere near the term's end, as I remember, our teacher mentioned what sounded like the most important result of them all—the fundamental theorem of algebra. Anything with a name like that, I figured, must be (for want of a better term) fundamental . Unfortunately, the teacher informed us that this theorem was much too advanced to state, let alone to investigate, at our current level of mathematical development. Fine. I was willing to wait. However, second-year algebra came and went, yet the fundamental theorem occupied only an obscure footnote from which I learned that it had something to do with factoring polynomials and solving polynomial equations. My semester in college algebra/precalculus the following year went a bit further, and I emerged vaguely aware that the fundamental theorem of algebra said that n th-degree polynomials could be factored into n (possibly complex) linear factors, and thus n th-degree polynomial equations must have n (possibly complex and possibly repeated) solutions. Of course, to that point we had done little with complex numbers and less with complex solutions of polynomial equations, so the whole business remained obscure and mysterious. Even in those pre-Watergate days, I began to sense that the mathematical establishment was engaged in some kind of cover-up to keep us ignorant of the true state of algebraic affairs.

FREE INTERPOLATION IN THE SPACES $ C^A_{r,\omega}$

Let the integer and the modulus of continuity be fixed, and let be the class of all functions continuous on the closed unit disk , analytic on its interior , and having an -continuous th derivative on . Consider for each and each fixed the polynomial in (the st partial sum of the Taylor series of in a neighborhood of ). Then for any two points (1.1)Let be a closed subset of . This article contains a solution of the problem of free interpolation in , formulated as follows: find necessary and sufficient conditions on such that for each collection of th-degree polynomials satisfying conditions of the type (1.1) for all there is a function with .Bibliography: 13 titles.

Further study of the linear regulator with disturbances--The case of vector disturbances satisfying a linear differential equation

In a previous paper [1], the conventional optimal linear regulator theory was extended to accommodate the case of external input disturbances \omega(t) which are not directly measurable but which can be assumed to satisfy d^{m+1}\omega(t)/dt^{m+1} = 0 , i.e., represented as m th-degree polynomials in time t with unknown coefficients. In this way, the optimal controller u^{0}(t) was obtained as the sum of: 1) a linear combination of the state variables x_{i}, i = 1,2,...,n , plus 2) a linear combination of the first (m + 1) time integrals of certain other linear combinations of the state variables. In the present paper, the results obtained in [1] are generalized to accommodate the case of unmeasurable disturbances \omega(t) which are known only to satisfy a given \rho th-degree linear differential equation D: d^{\rho}\omega(t)/dt^{\rho} + \beta_{\rho}d^{\rho-1}\omega(t)/dt^{\rho-1}+...+\beta_{2}d\omega/dt + \beta_{1}\omega=0 where the coefficients \beta_{i}, i = 1,...,\rho , are known. By this means, a dynamical feedback controller is derived which will consistently maintain state regulation x(t) \approx 0 in the face of any and every external disturbance function \omega(t) which satisfies the given differential equation D -even steady-state periodic or unstable functions \omega(t) . An essentially different method of deriving this result, based on stabilization theory, is also described, In each cases the results are extended to the case of vector control and vector disturbance.

A note on control systems with one nonlinear element

An important class of nonlinear control systems can be represented by a single-loop negative feedback circuit having one isolated, instantaneous type, nonlinear element in series with a linear transfer function with m th-degree numerator polynomial and n th-degree denominator polynomial ( m \leq n ). In the literature on automatic control, many of the techniques and theorems associated with this class of systems are restricted in application to the special case m . In this short paper, a technique is described whereby the case m = n can be transformed to the case m . By this means, some of the results that heretofore were restricted to the case m can be successfully applied to the case m = n . In particular, this procedure, together with a certain theorem about nonlinear functions with feedback, makes it possible to apply some recent results in absolute stability to certain cases of m = n .