Derivative Of Polynomial Research Articles

Application of Bernstein collocation method for solving the generalized regularized long wave equations

Abstract The regularized and the modified regularized long wave (RLW and MRLW) equations are solved numerically by the Bernstein polynomials in both the space and time directions based on Kronecker product. In this paper, we applied a fully different Bernstein collocation method than the other methods which used Bernstein polynomials to solve the problems. The approximate solution is defined by the Bernstein polynomials in all directions. A general form for any m derivative of any Bernstein polynomials is constructed. A general matrix form for the vector of any m derivative of any Bernstein polynomials is also constructed. Convergence study for the proposed numerical scheme is investigated. To determine the conservation properties of the RLW and MRLW equations, three invariants of motion ( I 1 , I 2 and I 3 ) are computed. To test the accuracy, two error norms ( ‖ E ‖ 2 and ‖ E ‖ ∞ ) are evaluated. Numerical outcomes and comparisons with other techniques for the single and the interaction of two solitary waves for RLW and MRLW equations are presented.

Some algebraic and analytic properties of generalized tribonacci and generalized tribonacci-Lucas like polynomials

In this study, we give another formula to all the generalized tribonacci [Formula: see text] and generalized tribonacci-Lucas [Formula: see text] polynomials by using the combinatorial calculus and their sum. Then, we give the explicit formula of partial derivative of such polynomials [Formula: see text], [Formula: see text] with respect to one of those variables and describe some properties.

Bernstein Type Inequalities for Polar Derivative of Polynomial

If is a polynomial of degree n such that in , , then Govil [ Proc. Nat. Acad. Sci., Vol. 50, pp. 50-52, 1980. ] proved 
 provided and attain their maxima at the same point on the circle , where .
 Equality in the above inequality holds for.
 In this paper, we extend the above inequality and an improved version of this into polar derivative of a polynomial.

Zeros of derivatives of Gaussian random polynomials on plane domains

In this paper, we study the expected density of zeros of derivatives of Gaussian random polynomials on plane domains of C. The Gaussian random polynomials are of the form RN=∑n=0Nanpn where an are i.i.d. complex Gaussian random variables and {pn} are orthonormal polynomials on the boundary of a bounded simply connected analytic plane domain relative to a weight. The main result is that the expected density of zeros of any k-th derivatives of RN tends to the equilibrium measure of the domain.

Extreme values of the derivative of Blaschke products and hypergeometric polynomials

A finite Blaschke product, restricted to the unit circle, is a smooth covering map. The maximum and minimum values of the derivative of this map reflect the geometry of the Blaschke product. We identify two classes of extremal Blaschke products: those that maximize the difference between the maximum and minimum of the derivative, and those that minimize it. Both classes turn out to have the same algebraic structure, being the quotient of two hypergeometric polynomials.

Hypergeometric fractional derivatives formula of shifted Chebyshev polynomials: tau algorithm for a type of fractional delay differential equations

Abstract This paper presents an explicit formula that approximates the fractional derivatives of Chebyshev polynomials of the first-kind in the Caputo sense. The new expression is given in terms of a terminating hypergeometric function of the type 4 F 3(1). The integer derivatives of Chebyshev polynomials of the first-kind are deduced as a special case of the fractional ones. The formula will be applied for obtaining a spectral solution of a certain type of fractional delay differential equations with the aid of an explicit Chebyshev tau method. The shifted Chebyshev polynomials of the first-kind are selected as basis functions and the spectral tau method is employed for obtaining the desired approximate solutions. The convergence and error analysis are discussed. Numerical results are presented illustrating the efficiency and accuracy of the proposed algorithm.

A Semicircle Law for Derivatives of Random Polynomials

Abstract Let $x_1, \dots , x_n$ be $n$ independent and identically distributed real-valued random variables with mean zero, unit variance, and finite moments of all remaining orders. We study the random polynomial $p_n$ having roots at $x_1, \dots , x_n$. We prove that for $\ell \in \mathbb{N}$ fixed as $n \rightarrow \infty $, the $(n-\ell )-$th derivative of $p_n^{}$ behaves like a Hermite polynomial: for $x$ in a compact interval, a suitable rescaling of $p_n^{(n-\ell )}$ starts behaving like the $\ell -$th probabilists’ Hermite polynomial subject to a random shift. Thus, there is a universality phenomenon when differentiating a random polynomial many times: the remaining roots follow a Wigner semicircle distribution.

New formulas of the high‐order derivatives of fifth‐kind Chebyshev polynomials: Spectral solution of the convection–diffusion equation

AbstractThis paper is dedicated to deriving novel formulae for the high‐order derivatives of Chebyshev polynomials of the fifth‐kind. The high‐order derivatives of these polynomials are expressed in terms of their original polynomials. The derived formulae contain certain terminating 4F3(1) hypergeometric functions. We show that the resulting hypergeometric functions can be reduced in the case of the first derivative. As an important application—and based on the derived formulas—a spectral tau algorithm is implemented and analyzed for numerically solving the convection–diffusion equation. The convergence and error analysis of the suggested double expansion is investigated assuming that the solution of the problem is separable. Some illustrative examples are presented to check the applicability and accuracy of our proposed algorithm.

Clique polynomials of $2$-connected $K_{5}$-free chordal graphs

In this paper, we give a generalization of the author's previous result about real rootedness of clique polynomials of connected $K_{4}$-free chordal graphs to the class of $2$-connected $K_{5}$-free chordal graphs. The main idea is based on the graph-theoretical interpretation of the second derivative of clique polynomials. Finally, we conclude the paper with several interesting open questions and conjectures.

Approximation by simple partial fractions in unbounded domains

For unbounded simply connected domains in the complex plane, bounded by several simple curves with regular asymptotic behaviour at infinity, we obtain necessary conditions and sufficient conditions for simple partial fractions (logarithmic derivatives of polynomials) with poles on the boundary of to be dense in the space of holomorphic functions in (with the topology of uniform convergence on compact subsets of ). In the case of a strip bounded by two parallel lines, we give estimates for the convergence rate to zero in the interior of of simple partial fractions with poles on the boundary of and with one fixed pole. Bibliography: 16 titles.

Some Results on a Two Variables Pell Polynomials

In this study, Pell polynomials in two variables, and their properties are investigated. Some formulas for two variables Pell polynomials are derived by matrices. By defining special formula for Pell polynomials in one variable, new important properties of Pell polynomials in two variables can be enabled to derive. A new exact formula expressing the partial derivatives of Pell polynomials explicitly of any degree in terms of Pell polynomials themselves is proved. A novel explicit formula, which constructs the two explicit formulas, which construct the two-dimension Pell polynomials expansion coefficients of a first partial derivative of a differentiable function in terms of their original expansion coefficients, is also included in the present article. The main advantage of the presented formulas is that the new properties of Pell polynomials in two variables greatly simplify the original problems and the result will lead to easy calculation of the coefficients of expansion. A direct spectral method along with the presented two variables Pell polynomials is proposed to solve the partial differential equation. Illustrated examples are included to demonstrate the validity of the technique.

Lp-inequalities for the polar derivative of a lacunary-type polynomial

In this paper, we extend an inequality concerning the polar derivative of a polynomial in Lp-norm to the class of lacunary polynomials and thereby obtain a bound that depends on some of the coefficients of the polynomial as well.

RETRACTED: On the derivative of a rational polynomial with prescribed poles

RETRACTED: On the derivative of a rational polynomial with prescribed poles

Energy management optimal strategy of FCHEV based on the Radau Pseudospectral method

Energy management of the fuel cell hybrid electric vehicles (FCHEVs) is vital with respect to improving FCHEV's performance and durability. Radau Pseudospectral method (RPM)-based optimal control of energy management of FCHEV is proposed to optimise the fuel cell's lifetime by means of reducing its performance degradation. To utilise the RPM, both state and control variables are approximated by global interpolation polynomials, and differential equations of state variables are approximated by the derivatives of interpolation polynomials. Accordingly, the optimal control problem is transformed into nonlinear problem to be solved. The fuel cell's performance degradation is selected as objective function. The optimal results in NEDC show that battery with larger capacity is more beneficial than smaller one for reducing the fuel cell's performance degradation, with the total time of large load change of the fuel cell reducing. The RPM is also an effective way to optimise other objectives to energy management.

Generalizations of some Zygmund-type integral inequalities for polar derivatives of a complex polynomial

We prove some results for algebraic polynomials in the complex plane that relate the L-norm of the polar derivative of a complex polynomial and the polynomial under some conditions. The obtained results include several interesting generalizations of some Zygmund-type integral inequalities for polynomials and derive polar derivative analogues of some classical Bernsteintype inequalities for the sup-norms on the unit disk as well.

Energy management optimal strategy of FCHEV based on the Radau Pseudospectral method

Energy management of the fuel cell hybrid electric vehicles (FCHEVs) is vital with respect to improving FCHEV's performance and durability. Radau Pseudospectral method (RPM)-based optimal control of energy management of FCHEV is proposed to optimise the fuel cell's lifetime by means of reducing its performance degradation. To utilise the RPM, both state and control variables are approximated by global interpolation polynomials, and differential equations of state variables are approximated by the derivatives of interpolation polynomials. Accordingly, the optimal control problem is transformed into nonlinear problem to be solved. The fuel cell's performance degradation is selected as objective function. The optimal results in NEDC show that battery with larger capacity is more beneficial than smaller one for reducing the fuel cell's performance degradation, with the total time of large load change of the fuel cell reducing. The RPM is also an effective way to optimise other objectives to energy management.

Generalization of Zygmund type inequalities for the sth derivative of polynomials

If p(z) is a polynomial of degree n and p(z) ≠ 0 in |z| < 1, it was proved by Hans and Lal [Anal. Math. 40, 105-115(2014)] that for any β ∈ C with |β| ≤ 1,1 ≤ s ≤ n, In this paper, we prove an inequality which gives an improved and generalized extension of the above inequality into Lγ norm.

Inequalities for the derivative of polynomials with restricted zeros

For a polynomial $\mathit{P(z)=\sum_{\nu =0}^{n}a_{\nu}z^{\nu}}$ of degree $\mathit{n}$ having all its zeros in $\mathit{|z|\leq k,k \geq 1}$, it was shown by Rather and Dar \cite{1} that $$\max_{|z|=1} |P^{\prime}(z)|\geq \frac{1}{1+k^n}\bigg(n+\frac{k^n|a_n|-|a_0|}{k^n|a_n|+|a_0|}\bigg)\max_{|z|=1}|P(z)|.$$ In this paper, we shall obtain some sharp estimates, which not only refine the above inequality but also generalize some well known Tur\'{a}n-type inequalities.

INEQUALITIES FOR ALGEBRAIC POLYNOMIALS ON AN ELLIPSE

The paper presents new solutions to two classical problems of approximation theory. The first problem is to find the polynomial that deviates least from zero on an ellipse. The second one is to find the exact upper bound of the uniform norm on an ellipse with foci \(\pm 1\) of the derivative of an algebraic polynomial with real coefficients normalized on the segment \([- 1,1]\).

Multiplicator type operators and approximation of periodic functions of one variable by trigonometric polynomials

The norms of the images of multiplier type operators generated by an arbitrary generator are estimated in terms of the best approximations of univariate periodic functions by trigonometric polynomials in the -spaces, . As corollaries, estimates for the quality of approximation by Fourier means, an inverse theorem of approximation theory, comparison theorems, an analogue of the Marchaud inequality for generalized moduli of smoothness defined by a periodic generator, as well as some constructive sufficient conditions for generalized smoothness and Bernstein type inequalities for generalized derivatives of trigonometric polynomials are obtained. Bibliography: 49 titles.