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ON SIMULTANEOUS APPROXIMATION TO A DIFFERENTIABLE FUNCTION AND ITS DERIVATIVE BY INVERSE PAL-TYPE INTERPOLATION POLYNOMIALS
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作者 Bao Yongguang (Hangzhou University, China) 《Analysis in Theory and Applications》 1995年第4期15-23,共9页
Let ξn-1<ξn-2 <ξn-2 <… < ξ1 be the zeros of the the (n -1)-th Legendre polynomial Pn-1(x) and - 1 = xn < xn-1 <… < x1 = 1 the zeros of the polynomial W n(x) =- n(n - 1) Pn-1(t)dt = (1 -x2)P&... Let ξn-1<ξn-2 <ξn-2 <… < ξ1 be the zeros of the the (n -1)-th Legendre polynomial Pn-1(x) and - 1 = xn < xn-1 <… < x1 = 1 the zeros of the polynomial W n(x) =- n(n - 1) Pn-1(t)dt = (1 -x2)P'n-1(x). By the theory of the inverse Pal-Type interpolation, for a function f(x) ∈ C[-1 1], there exists a unique polynomial Rn(x) of degree 2n - 2 (if n is even) satisfying conditions Rn(f,ξk) = f(∈ek)(1≤ k≤ n - 1) ;R'n(f,xk) = f'(xk)(1≤ k≤ n). This paper discusses the simultaneous approximation to a differentiable function f by inverse Pal-Type interpolation polynomial {Rn(f,x)} (n is even) and the main result of this paper is that if f ∈ C'[1,1], r≥2, n≥ + 2> and n is even thenholds uniformly for all x ∈ [- 1,1], where h(x) = 1 + 展开更多
关键词 MATH In ON SIMULTANEOUS APPROXIMATION TO A DIFFERENTIABLE FUNCTION AND ITS derivative BY INVERSE PAL-TYPE interpolation polynomialS PAL ITS
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Novel Investigation of Stochastic Fractional Differential Equations Measles Model via the White Noise and Global Derivative Operator Depending on Mittag-Leffler Kernel 被引量:1
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作者 Saima Rashid Fahd Jarad 《Computer Modeling in Engineering & Sciences》 SCIE EI 2024年第6期2289-2327,共39页
Because of the features involved with their varied kernels,differential operators relying on convolution formulations have been acknowledged as effective mathematical resources for modeling real-world issues.In this p... Because of the features involved with their varied kernels,differential operators relying on convolution formulations have been acknowledged as effective mathematical resources for modeling real-world issues.In this paper,we constructed a stochastic fractional framework of measles spreading mechanisms with dual medication immunization considering the exponential decay and Mittag-Leffler kernels.In this approach,the overall population was separated into five cohorts.Furthermore,the descriptive behavior of the system was investigated,including prerequisites for the positivity of solutions,invariant domain of the solution,presence and stability of equilibrium points,and sensitivity analysis.We included a stochastic element in every cohort and employed linear growth and Lipschitz criteria to show the existence and uniqueness of solutions.Several numerical simulations for various fractional orders and randomization intensities are illustrated. 展开更多
关键词 Measles epidemic model Atangana-Baleanu Caputo-Fabrizio differential operators existence and uniqueness qualitative analysis Newton interpolating polynomial
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Efficient Difference Schemes for the Caputo-Tempered Fractional Diffusion Equations Based on Polynomial Interpolation
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作者 Le Zhao Can Li Fengqun Zhao 《Communications on Applied Mathematics and Computation》 2021年第1期1-40,共40页
The tempered fractional calculus has been successfully applied for depicting the time evolution of a system describing non-Markovian diffusion particles.The related governing equations are a series of partial differen... The tempered fractional calculus has been successfully applied for depicting the time evolution of a system describing non-Markovian diffusion particles.The related governing equations are a series of partial differential equations with tempered fractional derivatives.Using the polynomial interpolation technique,in this paper,we present three efficient numerical formulas,namely the tempered L1 formula,the tempered L1-2 formula,and the tempered L2-1_(σ)formula,to approximate the Caputo-tempered fractional derivative of orderα∈(0,1).The truncation error of the tempered L1 formula is of order 2-α,and the tempered L1-2 formula and L2-1_(σ)formula are of order 3-α.As an application,we construct implicit schemes and implicit ADI schemes for one-dimensional and two-dimensional time-tempered fractional diffusion equations,respectively.Furthermore,the unconditional stability and convergence of two developed difference schemes with tempered L1 and L2-1_(σ)formulas are proved by the Fourier analysis method.Finally,we provide several numerical examples to demonstrate the correctness and effectiveness of the theoretical analysis. 展开更多
关键词 Caputo-tempered fractional derivative polynomial interpolation Implicit ADI schemes STABILITY
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Null subcarriers based Doppler scale estimation with polynomial interpolation for multicarrier communication over ultrawideband underwater acoustic channels 被引量:1
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作者 Yang Chen Jingwei Yin +2 位作者 Ling Zou Dan Yang Yuan Cao 《Journal of Systems Engineering and Electronics》 SCIE EI CSCD 2015年第6期1177-1183,共7页
This paper addresses the extremal problem of the null subcarriers based Doppler scale estimation in underwater acoustic (UWA) orthogonal frequency division multiplexing (OFDM) communication. The cost function cons... This paper addresses the extremal problem of the null subcarriers based Doppler scale estimation in underwater acoustic (UWA) orthogonal frequency division multiplexing (OFDM) communication. The cost function constructed of the total energy of null subcarriers through discrete Fourier transform (DFT) is proposed. The frequencies of null subcarriers are identified from non-uniform Doppler shift at each tentative scaling factor. Then it is proved that the cost function can be fitted as a quadratic polynomial near the global minimum. An accurate Doppler scale estimation is achieved by the location of the global scarifying precision and increasing the computation minimum through polynomial interpolation, without complexity. A shallow water experiment is conducted to demonstrate the performance of the proposed method. Excellent performance results are obtained in ultrawideband UWA channels with a relative bandwidth of 67%, when the transmitter and the receiver are moving at a relative speed of 5 kn, which validates the proposed method. 展开更多
关键词 underwater acoustic (UWA) communication orthogo-nal frequency division multiplexing (OFDM) Doppler scale estimation polynomial interpolation.
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SIMULTANEOUSE APPROXIMATION TO A DIFFERENTIABLE FUNCTION AND ITS DERIVATIVES BY LAGRANGE INTERPOLATING POLYNOMIALS 被引量:1
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作者 T.F.Xie S.P.Zhou 《Analysis in Theory and Applications》 1994年第4期100-109,共10页
This paper establishes the following pointwise result for simultancous Lagrange imterpolating approxima- tion:,then |f^(k)(x)-P_n^(k)(f,x)|=O(1)△_n^(q-k)(x)ω where P_n(f,x)is the Lagrange interpolating potynomial of... This paper establishes the following pointwise result for simultancous Lagrange imterpolating approxima- tion:,then |f^(k)(x)-P_n^(k)(f,x)|=O(1)△_n^(q-k)(x)ω where P_n(f,x)is the Lagrange interpolating potynomial of deereeon the nodes X_nUY_n(see the definition of the next). 展开更多
关键词 SIMULTANEOUSE APPROXIMATION TO A DIFFERENTIABLE FUNCTION AND ITS derivativeS BY LAGRANGE INTERPOLATING polynomialS APPI ZR
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Some Inequalities Concerning the Polar Derivative of a Polynomial-II 被引量:1
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作者 Abdullah Mir Bilal Dar 《Analysis in Theory and Applications》 2013年第4期384-389,共6页
In this paper, we consider the class of polynomials P(z)= anz^n+ ∑vn=μan-vz^n-v,1≤μ≤n , having all zeros in |z|≤k, k ≤1 and thereby present an alternative proof, independent of Laguerre's theorem, of an... In this paper, we consider the class of polynomials P(z)= anz^n+ ∑vn=μan-vz^n-v,1≤μ≤n , having all zeros in |z|≤k, k ≤1 and thereby present an alternative proof, independent of Laguerre's theorem, of an inequality concerning the polar derivative of a polynomial. 展开更多
关键词 Polar derivative of a polynomial.
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HAAR EXPANSIONS OF A CLASS OF FRACTAL INTERPOLATION FUNCTIONS AND THEIR LOGICAL DERIVATIVES 被引量:1
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作者 Sha Zhen Chen Gang Zhejiang University,China 《Analysis in Theory and Applications》 1993年第4期73-88,共16页
In this paper,we study a special class of fractal interpolation functions,and give their Haar-wavelet expansions.On the basis of the expansions,we investigate the H(o|¨)lder smoothness of such functions and their... In this paper,we study a special class of fractal interpolation functions,and give their Haar-wavelet expansions.On the basis of the expansions,we investigate the H(o|¨)lder smoothness of such functions and their logical derivatives of order α. 展开更多
关键词 HAAR EXPANSIONS OF A CLASS OF FRACTAL interpolation FUNCTIONS AND THEIR LOGICAL derivativeS der HAAR FIF
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ON THE POINTWISE ESTIMATIONS OF APPROXIMATION OF FUNCTIONS AND THEIR DERIVATIVES BY HERMITE INTERPOLATION 被引量:1
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作者 Tingfan Xie Ziyu Wang China Institute of Metrology, China Henan University, China 《Analysis in Theory and Applications》 1994年第3期45-55,共11页
The object of this paper is to establish the pointwise estimations of approximation of functions in C^1 and their derivatives by Hermite interpolation polynomials. The given orders have been proved to be exact in gen-... The object of this paper is to establish the pointwise estimations of approximation of functions in C^1 and their derivatives by Hermite interpolation polynomials. The given orders have been proved to be exact in gen- eral. 展开更多
关键词 MATH ON THE POINTWISE ESTIMATIONS OF APPROXIMATION OF FUNCTIONS AND THEIR derivativeS BY HERMITE interpolation 石瓦
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Study of Fractional Order Dynamical System of Viral Infection Disease under Piecewise Derivative 被引量:2
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作者 Kamal Shah Hafsa Naz +1 位作者 Thabet Abdeljawad Bahaaeldin Abdalla 《Computer Modeling in Engineering & Sciences》 SCIE EI 2023年第7期921-941,共21页
This research aims to understand the fractional order dynamics of the deadly Nipah virus(NiV)disease.We focus on using piecewise derivatives in the context of classical and singular kernels of power operators in the C... This research aims to understand the fractional order dynamics of the deadly Nipah virus(NiV)disease.We focus on using piecewise derivatives in the context of classical and singular kernels of power operators in the Caputo sense to investigate the crossover behavior of the considered dynamical system.We establish some qualitative results about the existence and uniqueness of the solution to the proposed problem.By utilizing the Newtonian polynomials interpolation technique,we recall a powerful algorithm to interpret the numerical findings for the aforesaid model.Here,we remark that the said viral infection is caused by an RNA type virus which can transmit from animals and also from an infected person to person.Fruits bats which are also known as flying foxes are one of the sources of transmission of NiV disease.Here in this work,we investigate its transmission mechanism through some new concepts of fractional calculus for further analysis and prediction.We present the approximate results for different compartments using different fractional orders.By using the piecewise derivative concept,we detect the crossover ormulti-steps behavior in the transmission dynamics of the mentioned disease.Therefore,the considered form of the derivative is used to deal with problems exhibiting crossover behaviors. 展开更多
关键词 NiV disease fractional calculus piecewise derivative qualitative results newton polynomial RNA virus
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ON EXTREMAL PROPERTIES FOR THE POLAR DERIVATIVE OF POLYNOMIALS
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作者 K. K. Dewan Arty Ahuja 《Analysis in Theory and Applications》 2011年第2期150-157,共8页
If p(z) is a polynomial of degree n having all its zeros on |z| = k, k ≤ 1, then it is proved[5] that In this paper, we generalize the above inequality by extending it to the polar derivative of a polynomial of t... If p(z) is a polynomial of degree n having all its zeros on |z| = k, k ≤ 1, then it is proved[5] that In this paper, we generalize the above inequality by extending it to the polar derivative of a polynomial of the type We also obtain certain new inequalities concerning the maximum modulus of a polynomial with restricted zeros. 展开更多
关键词 polynomial ZEROS INEQUALITY polar derivative
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SOME INTEGRAL INEQUALITIES FOR THE POLAR DERIVATIVE OF A POLYNOMIAL
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作者 Abdullah Mir Sajad Amin Baba 《Analysis in Theory and Applications》 2011年第4期340-350,共11页
If P(z) is a polynomial of degree n which does not vanish in |z| 〈 1, then it is recently proved by Rather [Jour. Ineq. Pure andAppl. Math., 9 (2008), Issue 4, Art. 103] that for every γ 〉 0 and every real or... If P(z) is a polynomial of degree n which does not vanish in |z| 〈 1, then it is recently proved by Rather [Jour. Ineq. Pure andAppl. Math., 9 (2008), Issue 4, Art. 103] that for every γ 〉 0 and every real or complex number a with | α | ≥ 1,{∫ 2π 0|DαP(e^iθ)|γdθ|}^1/γ≤n(|α|+1)Cγ{∫2π0|P(e^iθ)|γ^dθ}^1/γ,Cγ={1/2π∫2π 0|1+e^iβ|^γdβ}^-1/γ,where DaP(z) denotes the polar derivative of P(z) with respect to α. In this paper we prove a result which not only provides a refinement of the above inequality but also gives a result of Aziz and Dawood [J. Approx. Theory, 54 (1988), 306-313] as a special case. 展开更多
关键词 polar derivative polynomial Zygmund inequality ZEROS
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Some Results on the Polar Derivative of a Polynomial
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作者 Abdullah Mir Bilal Dar 《Analysis in Theory and Applications》 2014年第3期306-317,共12页
Let P(z) be a polynomial of degree n and for any complex number α, let DαP(z) = nP(z)+ (α-z)P' (z) denote the polar derivative of P(z) with respect to α. In this paper, we obtain certain inequalities... Let P(z) be a polynomial of degree n and for any complex number α, let DαP(z) = nP(z)+ (α-z)P' (z) denote the polar derivative of P(z) with respect to α. In this paper, we obtain certain inequalities for the polar derivative of a polynomial with restricted zeros. Our results generalize and sharpen some well-known polynomial inequalities. 展开更多
关键词 polynomial ZEROS polar derivative Bemstein inequality.
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Full-vectorial analysis of optical waveguides by the finite difference method based on polynomial interpolation
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作者 肖金标 张明德 孙小菡 《Chinese Physics B》 SCIE EI CAS CSCD 2006年第1期143-148,共6页
Based on the polynomial interpolation, a new finite difference (FD) method in solving the full-vectorial guidedmodes for step-index optical waveguides is proposed. The discontinuities of the normal components of the... Based on the polynomial interpolation, a new finite difference (FD) method in solving the full-vectorial guidedmodes for step-index optical waveguides is proposed. The discontinuities of the normal components of the electric field across abrupt dielectric interfaces are considered in the absence of the limitations of scalar and semivectorial approximation, and the present PD scheme can be applied to both uniform and non-uniform mesh grids. The modal propagation constants and field distributions for buried rectangular waveguides and optical rib waveguides are presented. The hybrid nature of the vectorial modes is demonstrated and the singular behaviours of the minor field components in the corners are observed. Moreover, solutions are in good agreement with those published early, which tests the validity of the present approach. 展开更多
关键词 polynomial interpolation finite difference full vectorial mode solver optical waveguides photonic integrated circuits
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Some Lγ Inequalities for the Polar Derivative of a Polynomial
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作者 Abdullah Mir M. Bidkham Bilal Dar 《Analysis in Theory and Applications》 CSCD 2017年第1期1-10,共10页
In this paper, we consider an operator Da which maps a polynomial P(z) in to DaP(z):= np(z)+ (a-z)P'(z), where and obtain some Lγ inequalities for lucanary polynomials having zeros in /z/ 〈 k 〈 1. Ou... In this paper, we consider an operator Da which maps a polynomial P(z) in to DaP(z):= np(z)+ (a-z)P'(z), where and obtain some Lγ inequalities for lucanary polynomials having zeros in /z/ 〈 k 〈 1. Our results yields several generaliza- tions and refinements of many known results and also provide an alternative proof of a result due to Dewan et al. [7], which is independent of Laguerre's theorem. 展开更多
关键词 Polar derivative polynomialS Lγ-inequalities in the complex domain Laguerre'stheorem.
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Inequalities for the Polar Derivatives of a Polynomial
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作者 B.A.Zargar 《Analysis in Theory and Applications》 2014年第4期425-432,共8页
Let P(z) be a polynomial of degree n, having all its zeros in |z|≤ 1. In this paper, we estimate kth polar derivative of P(z) on |z| = 1 and thereby obtain compact generalizations of some known results which ... Let P(z) be a polynomial of degree n, having all its zeros in |z|≤ 1. In this paper, we estimate kth polar derivative of P(z) on |z| = 1 and thereby obtain compact generalizations of some known results which among other things yields a refinement of a result due to Paul Tura'n. 展开更多
关键词 Polar derivative of a polynomial maximum modulus Bernstiens inequality.
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Complex derivatives valuation: applying the Least-Squares Monte Carlo Simulation Method with several polynomial basis
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作者 Ursula Silveira Monteiro de Lima Carlos Patricio Samanez 《Financial Innovation》 2016年第1期1-14,共14页
Background:This article investigates the Least-Squares Monte Carlo Method by using different polynomial basis in American Asian Options pricing.The standard approach in the option pricing literature is to choose the b... Background:This article investigates the Least-Squares Monte Carlo Method by using different polynomial basis in American Asian Options pricing.The standard approach in the option pricing literature is to choose the basis arbitrarily.By comparing four different polynomial basis we show that the choice of basis interferes in the option's price.Methods:We assess Least-Squares Method performance in pricing four different American Asian Options by using four polynomial basis:Power,Laguerre,Legendre and Hermite A.To every American Asian Option priced,three sets of parameters are used in order to evaluate it properly.Results:We show that the choice of the basis interferes in the option's price by showing that one of them converges to the option's value faster than any other by using fewer simulated paths.In the case of an Amerasian call option,for example,we find that the preferable polynomial basis is Hermite A.For an Amerasian put option,the Power polynomial basis is recommended.Such empirical outcome is theoretically unpredictable,since in principle all basis can be indistinctly used when pricing the derivative.Conclusion:In this article The Least-Squares Monte Carlo Method performance is assessed in pricing four different types of American Asian Options by using four different polynomial basis through three different sets of parameters.Our results suggest that one polynomial basis is best suited to perform the method when pricing an American Asian option.Theoretically all basis can be indistinctly used when pricing the derivative.However,our results does not confirm these.We find that when pricing an American Asian put option,Power A is better than the other basis we have studied here whereas when pricing an American Asian call,Hermite A is better. 展开更多
关键词 Complex derivatives valuation Least-Squares Monte Carlo Method Amerasian options polynomial basis
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ON APPROXIMATION OF A FUNCTION AND ITS DERIVATIVES BY A POLYNOMIAL AND ITS DERIVATIVES
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作者 T.Kilgore J.Szabados 《Analysis in Theory and Applications》 1994年第3期93-103,共11页
Let g∈C^q[-1, 1] be such that g^((k))(±1)=0 for k=0,…,q. Let P_n be an algebraic polynomial of degree at most n, such that P_n^((k))(±1)=0 for k=0,…,[_2~ (q+1)]. Then P_n and its derivatives P_n^((k)) fo... Let g∈C^q[-1, 1] be such that g^((k))(±1)=0 for k=0,…,q. Let P_n be an algebraic polynomial of degree at most n, such that P_n^((k))(±1)=0 for k=0,…,[_2~ (q+1)]. Then P_n and its derivatives P_n^((k)) for k≤q well approximate g and its respective derivatives, provided only that P_n well approxi- mates g itself in the weighted norm ‖g(x)-P_n(x) (1-x^2)^(1/2)~q‖ This result is easily extended to an arbitrary f∈C^q[-1, 1], by subtracting from f the polynomial of minnimal degree which interpolates f^((0))…,f^((q)) at±1. As well as providing easy criteria for judging the simultaneous approximation properties of a given Polynomial to a given function, our results further explain the similarities and differences between algebraic polynomial approximation in C^q[-1, 1] and trigonometric polynomial approximation in the space of q times differentiable 2π-periodic functions. Our proofs are elementary and basic in character, permitting the construction of actual error estimates for simultaneous approximation proedures for small values of q. 展开更多
关键词 ON APPROXIMATION OF A FUNCTION AND ITS derivativeS BY A polynomial AND ITS derivativeS ITS
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A NEW DERIVATIVE FREE OPTIMIZATION METHOD BASED ON CONIC INTERPOLATION MODEL 被引量:9
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作者 倪勤 胡书华 《Acta Mathematica Scientia》 SCIE CSCD 2004年第2期281-290,共10页
In this paper, a new derivative free trust region method is developed based on the conic interpolation model for the unconstrained optimization. The conic interpolation model is built by means of the quadratic model f... In this paper, a new derivative free trust region method is developed based on the conic interpolation model for the unconstrained optimization. The conic interpolation model is built by means of the quadratic model function, the collinear scaling formula, quadratic approximation and interpolation. All the parameters in this model are determined by objective function interpolation condition. A new derivative free method is developed based upon this model and the global convergence of this new method is proved without any information on gradient. 展开更多
关键词 derivative free optimization method conic interpolation model quadratic interpolation model trust region method unconstrained optimization
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Polar Derivative Versions of Polynomial Inequalities
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作者 Barchand Chanam 《Advances in Pure Mathematics》 2015年第12期745-755,共11页
Let be a polynomial of degree n and for a complex number , let ?denote the polar derivative of the polynomial ?with respect to . In this paper, first we extend as well as generalize the result proved by Dewan and Mir ... Let be a polynomial of degree n and for a complex number , let ?denote the polar derivative of the polynomial ?with respect to . In this paper, first we extend as well as generalize the result proved by Dewan and Mir [Inter. Jour. Math. and Math. Sci., 16 (2005), 2641-2645] to polar derivative. Besides, another result due to Dewan et al. [J. Math. Anal. Appl. 269 (2002), 489-499] is also extended to polar derivative. 展开更多
关键词 polynomialS POLAR derivative of a polynomial ZEROS EXTREMAL polynomialS
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Some Inequalities on Polar Derivative of Polynomial Having No Zero in a Disc
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作者 Barchand Chanam 《Advances in Pure Mathematics》 2015年第13期796-803,共8页
Let , , be a polynomial of degree n having no zero in , , then Qazi [Proc. Amer. Math. Soc., 115 (1992), 337-343] proved . In this paper, we first extend the above inequality to polar derivative of a polynomial. Furth... Let , , be a polynomial of degree n having no zero in , , then Qazi [Proc. Amer. Math. Soc., 115 (1992), 337-343] proved . In this paper, we first extend the above inequality to polar derivative of a polynomial. Further, as an application of our result, we extend a result due to Dewan et al. [Southeast Asian Bull. Math., 27 (2003), 591-597] to polar derivative. 展开更多
关键词 polynomialS INEQUALITIES POLAR derivative of a polynomial ZEROS
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