Duality between Bessel Functions and Chebyshev Polynomials in Expansions of Functions

In expansions of arbitrary functions in Bessel functions or Spherical Bessel functions, a dual partner set of polynomials play a role. For the Bessel functions, these are the Chebyshev polynomials of first kind and for the Spherical Bessel functions the Legendre polynomials. These two sets of functions appear in many formulas of the expansion and in the completeness and (bi)-orthogonality relations. The analogy to expansions of functions in Taylor series and in moment series and to expansions in Hermite functions is elaborated. Besides other special expansion, we find the expansion of Bessel functions in Spherical Bessel functions and their inversion and of Chebyshev polynomials of first kind in Legendre polynomials and their inversion. For the operators which generate the Spherical Bessel functions from a basic Spherical Bessel function, the normally ordered (or disentangled) form is found.

Research on Low Sampling Rate Digital Pre-distortion Technology Based on Improved Chebyshev Polynomial

This paper presents a low sampling rate digital pre-distortion technique based on an improved Chebyshev polynomial for the non-linear distortion problem of amplifiers in 5G broadband communication systems.An improved Chebyshev polynomial is used to construct the behavioural model of the broadband amplifier,and an undersampling technique is used to sample the output signal of the amplifier,reduce the sampling rate,and extract the pre-distortion parameters from the sampled signal through an indirect learning structure to finally correct the non-linearity of the amplifier system.This technique is able to improve the linearity and efficiency of the power amplifier and provides better flexibility.Experimental results show that by constructing the behavioural model of the amplifier using memory polynomials(MP),generalised polynomials(GMP)and modified Chebyshev polynomials respectively,the adjacent channel power ratio of the obtained system can be improved by more than 13.87d B,17.6dB and 19.98dB respectively compared to the output signal of the amplifier without digital pre-distortion.The Chebyshev polynomial improves the neighbourhood channel power ratio by 6.11dB and 2.38dB compared to the memory polynomial and generalised polynomial respectively,while the normalised mean square error is effectively improved and enhanced.This shows that the improved Chebyshev pre-distortion can guarantee the performance of the system and improve the non-linearity better.

Analysis of stochastic bifurcation and chaos in stochastic Duffing-van der Pol system via Chebyshev polynomial approximation

The Chebyshev polynomial approximation is applied to investigate the stochastic period-doubling bifurcation and chaos problems of a stochastic Duffing-van der Pol system with bounded random parameter of exponential probability density function subjected to a harmonic excitation. Firstly the stochastic system is reduced into its equivalent deterministic one, and then the responses of stochastic system can be obtained by numerical methods. Nonlinear dynamical behaviour related to stochastic period-doubling bifurcation and chaos in the stochastic system is explored. Numerical simulations show that similar to its counterpart in deterministic nonlinear system of stochastic period-doubling bifurcation and chaos may occur in the stochastic Duffing-van der Pol system even for weak intensity of random parameter. Simply increasing the intensity of the random parameter may result in the period-doubling bifurcation which is absent from the deterministic system.

Precise integration methods based on the Chebyshev polynomial of the first kind

This paper introduces two new types of precise integration methods based on Chebyshev polynomial of the first kind for dynamic response analysis of structures, namely the integral formula method （IFM） and the homogenized initial system method （HISM）. In both methods, nonlinear variable loadings within time intervals are simulated using Chebyshev polynomials of the first kind before a direct integration is performed. Developed on the basis of the integral formula, the recurrence relationship of the integral computation suggested in this paper is combined with the Crout decomposed method to solve linear algebraic equations. In this way, the IFM based on Chebyshev polynomial of the first kind is constructed. Transforming the non-homogenous initial system to the homogeneous dynamic system, and developing a special scheme without dimensional expansion, the HISM based on Chebyshev polynomial of the first kind is able to avoid the matrix inversion operation. The accuracy of the time integration schemes is examined and compared with other commonly used schemes, and it is shown that a greater accuracy as well as less time consuming can be achieved. Two numerical examples are presented to demonstrate the applicability of these new methods.

ON THE COEFFICIENTS OF DIFFERENTIATED EXPANSIONS AND DERIVATIVES OF CHEBYSHEV POLYNOMIALS OF THE THIRD AND FOURTH KINDS

Two new analytical formulae expressing explicitly the derivatives of Chebyshev polynomials of the third and fourth kinds of any degree and of any order in terms of Chebyshev polynomials of the third and fourth kinds themselves are proved. Two other explicit formulae which express the third and fourth kinds Chebyshev expansion coefficients of a general-order derivative of an infinitely differentiable function in terms of their original expansion coefficients are also given. Two new reduction formulae for summing some terminating hypergeometric functions of unit argument are deduced. As an application of how to use Chebyshev polynomials of the third and fourth kinds for solving high-order boundary value problems, two spectral Galerkin numerical solutions of a special linear twelfth-order boundary value problem are given.

Key Management Using Chebyshev Polynomials for Mobile Ad Hoc Networks

A dedicated key server cannot be instituted to manage keys for MANETs since they are dynamic and unstable. The Lagrange's polynomial and curve fitting are being used to implement hierarchical key management for Mobile Ad hoc Networks(MANETs). The polynomial interpolation by Lagrange and curve fitting requires high computational efforts for higher order polynomials and moreover they are susceptible to Runge's phenomenon. The Chebyshev polynomials are secure, accurate, and stable and there is no limit to the degree of the polynomials. The distributed key management is a big challenge in these time varying networks. In this work, the Chebyshev polynomials are used to perform key management and tested in various conditions. The secret key shares generation, symmetric key construction and key distribution by using Chebyshev polynomials are the main elements of this projected work. The significance property of Chebyshev polynomials is its recursive nature. The mobile nodes usually have less computational power and less memory, the key management by using Chebyshev polynomials reduces the burden of mobile nodes to implement the overall system.

Using Chebyshev polynomial interpolation to improve the computational efficiency of gravity models near an irregularly-shaped asteroid

In asteroid rendezvous missions, the dynamical environment near an asteroid’s surface should be made clear prior to launch of the mission. However, most asteroids have irregular shapes,which lower the efficiency of calculating their gravitational field by adopting the traditional polyhedral method. In this work, we propose a method to partition the space near an asteroid adaptively along three spherical coordinates and use Chebyshev polynomial interpolation to represent the gravitational acceleration in each cell. Moreover, we compare four different interpolation schemes to obtain the best precision with identical initial parameters. An error-adaptive octree division is combined to improve the interpolation precision near the surface. As an example, we take the typical irregularly-shaped nearEarth asteroid 4179 Toutatis to demonstrate the advantage of this method; as a result, we show that the efficiency can be increased by hundreds to thousands of times with our method. Our results indicate that this method can be applicable to other irregularly-shaped asteroids and can greatly improve the evaluation efficiency.

DENSITY OF MARKOV SYSTEMS AND ZEROS OF CHEBYSHEV POLYNOMIALS

We raise and partly answer the question: whether there exists a Markov system with respect to which the zeros of the Chebyshev polynomials are dense, but the maximum length of a zero free interval of the nth Chebyshev polynomial does not tends to zero. We also draw the conclu- tion that a Markov system, under an additional assumption, is dense if and only if the maxi- mum length of a zero free interval of the nth associated Chebyshev polynomial tends to zero.

Chebyshev Polynomial-Based Analytic Solution Algorithm with Efficiency, Stability and Sensitivity for Classic Vibrational Constant Coefficient Homogeneous IVPs with Derivative Orders n, n-1, n-2

The Chebyshev polynomials are harnessed as functions of the one parameter of the nondimensionalized differential equation for trinomial homogeneous linear differential equations of arbitrary order n that have constant coefficients and exhibit vibration. The use of the Chebyshev polynomials allows calculation of the analytic solutions for arbitrary n in terms of the orthogonal Chebyshev polynomials to provide a more stable solution form and natural sensitivity analysis in terms of one parameter and the initial conditions in 6n + 7 arithmetic operations and one square root.

Lucas Symbolic Formulae and Generating Functions for Chebyshev Polynomials

This work shows that each kind of Chebyshev polynomials may be calculated from a symbolic formula similar to the Lucas formula for Bernoulli polynomials. It exposes also a new approach for obtaining generating functions of them by operator calculus built from the derivative and the positional operators.

Asymptotic coefficients and errors for Chebyshev polynomial approximations with weak endpoint singularities: Effects of different bases

When one solves differential equations by a spectral method,it is often convenient to shift from Chebyshev polynomials Tn(x) with coefficients anto modified basis functions that incorporate the boundary conditions.For homogeneous Dirichlet boundary conditions,u(±1)=0,popular choices include the "Chebyshev difference basis" ζn(x)≡Tn+2(x)-Tn(x) with coefficients here denoted by bnand the "quadratic factor basis" Qn(x)≡(1-x2)Tn(x) with coefficients cn.If u(x) is weakly singular at the boundary,then the coefficients andecrease proportionally to O(A(n)/nκ) for some positive constant κ,where A(n) is a logarithm or a constant.We prove that the Chebyshev difference coefficients bndecrease more slowly by a factor of 1/n while the quadratic factor coefficients cndecrease more slowly still as O(A(n)/nκ-2).The error for the unconstrained Chebyshev series,truncated at degree n=N,is O(|A(N)|/Nκ) in the interior,but is worse by one power of N in narrow boundary layers near each of the endpoints.Despite having nearly identical error norms in interpolation,the error in the Chebyshev basis is concentrated in boundary layers near both endpoints,whereas the error in the quadratic factor and difference basis sets is nearly uniformly oscillating over the entire interval in x.Meanwhile,for Chebyshev polynomials,the values of their derivatives at the endpoints are O(n2),but only O(n) for the difference basis.Furthermore,we give the asymptotic coefficients and rigorous error estimates of the approximations in these three bases,solved by the least squares method.We also find an interesting fact that on the face of it,the aliasing error is regarded as a bad thing;actually,the error norm associated with the downward curving spectral coefficients decreases even faster than the error norm of infinite truncation.But the premise is under the same basis,and when involving different bases,it may not be established yet.

Fast algorithms of public key cryptosystem based on Chebyshev polynomials over finite field

The computation of Chebyshev polynomial over finite field is a dominating operation for a public key cryptosystem.Two generic algorithms with running time of have been presented for this computation：the matrix algorithm and the characteristic polynomial algorithm,which are feasible but not optimized.In this paper,these two algorithms are modified in procedure to get faster execution speed.The complexity of modified algorithms is still,but the number of required operations is reduced,so the execution speed is improved.Besides,a new algorithm relevant with eigenvalues of matrix in representation of Chebyshev polynomials is also presented,which can further reduce the running time of that computation if certain conditions are satisfied.Software implementations of these algorithms are realized,and the running time comparison is given.Finally an efficient scheme for the computation of Chebyshev polynomial over finite field is presented.

Robust Control of Robotic Manipulators in the Task-Space Using an Adaptive Observer Based on Chebyshev Polynomials

In this paper,an adaptive observer for robust control of robotic manipulators is proposed.The lumped uncertainty is estimated using Chebyshev polynomials.Usually,the uncertainty upper bound is required in designing observer-controller structures.However,obtaining this bound is a challenging task.To solve this problem,many uncertainty estimation techniques have been proposed in the literature based on neuro-fuzzy systems.As an alternative,in this paper,Chebyshev polynomials have been applied to uncertainty estimation due to their simpler structure and less computational load.Based on strictly-positive-rea Lyapunov theory,the stability of the closed-loop system can be verified.The Chebyshev coefficients are tuned based on the adaptation rules obtained in the stability analysis.Also,to compensate the truncation error of the Chebyshev polynomials,a continuous robust control term is designed while in previous related works,usually a discontinuous term is used.An SCARA manipulator actuated by permanent magnet DC motors is used for computer simulations.Simulation results reveal the superiority of the designed method.

Multilayer perceptron and Chebyshev polynomials-based functional link artificial neural network for solving differential equations

This paper discusses the issues of computational efforts and the accuracy of solutions of differential equations using multilayer perceptron and Chebyshev polynomials-based functional link artificial neural networks.Some ordinary and partial differential equations have been solved by both these techniques and pros and cons of both these type of feedforward networks have been discussed in detail.Apart from that,various factors that affect the accuracy of the solution have also been analyzed.

Efficient Algorithms for Approximating Particular Solutions of Elliptic Equations Using Chebyshev Polynomials

In this paper,we propose efficient algorithms for approximating particular solutions of second and fourth order elliptic equations.The approximation of the particular solution by a truncated series of Chebyshev polynomials and the satisfaction of the differential equation lead to upper triangular block systems,each block being an upper triangular system.These systems can be solved efficiently by standard techniques.Several numerical examples are presented for each case.

Efficient method to calculate the eigenvalues of the Zakharov–Shabat system

A numerical method is proposed to calculate the eigenvalues of the Zakharov–Shabat system based on Chebyshev polynomials. A mapping in the form of tanh(ax) is constructed according to the asymptotic of the potential function for the Zakharov–Shabat eigenvalue problem. The mapping can distribute Chebyshev nodes very well considering the gradient for the potential function. Using Chebyshev polynomials, tanh(ax) mapping, and Chebyshev nodes, the Zakharov–Shabat eigenvalue problem is transformed into a matrix eigenvalue problem. This method has good convergence for the Satsuma–Yajima potential and the convergence rate is faster than the Fourier collocation method. This method is not only suitable for simple potential functions but also converges quickly for a complex Y-shape potential. It can also be further extended to other linear eigenvalue problems.

Explicit Shifted Second-kind Chebyshev Spectral Treatment for Fractional Riccati Differential Equation

This paper is confined to analyzing and implementing new spectral solutions of the fractional Riccati differential equation based on the application of the spectral tau method.A new explicit formula for approximating the fractional derivatives of shifted Chebyshev polynomials of the second kind in terms of their original polynomials is established.This formula is expressed in terms of a certain terminating hypergeometric function of the type_(4)F_(3)(1).This hypergeometric function is reduced in case of the integer case into a certain terminating hypergeometric function of the type 3 F 2(1)which can be summed with the aid of Watson’s identity.Six illustrative examples are presented to ensure the applicability and accuracy of the proposed algorithm.

Chebyshev finite spectral method with extended moving grids

A Chebyshev finite spectral method on non-uniform meshes is proposed. An equidistribution scheme for two types of extended moving grids is used to generate grids. One type is designed to provide better resolution for the wave surface, and the other type is for highly variable gradients. The method has high-order accuracy because of the use of the Chebyshev polynomial as the basis function. The polynomial is used to interpolate the values between the two non-uniform meshes from a previous time step to the current time step. To attain high accuracy in the time discretization, the fourth-order Adams-Bashforth-Moulton predictor and corrector scheme is used. To avoid numerical oscillations caused by the dispersion term in the Korteweg-de Vries （KdV） equation, a numerical technique on non-uniform meshes is introduced. The proposed numerical scheme is validated by the applications to the Burgers equation （nonlinear convectiondiffusion problems） and the KdV equation （single solitary and 2-solitary wave problems）, where analytical solutions are available for comparisons. Numerical results agree very well with the corresponding analytical solutions in all cases.

Numerical solution to the Falkner-Skan equation:a novel numerical approach through the new rational α-polynomials

The new rationalα-polynomials are used to solve the Falkner-Skan equation.These polynomials are equipped with an auxiliary parameter.The approximated solution to the Falkner-Skan equation is obtained by the new rational a-polynomials with unknown coefficients.To find the unknown coefficients and the auxiliary parameter contained in the polynomials,the collocation method with Chebyshev-Gauss points is used.The numerical examples show the efficiency of this method.

AN ACCURATE SOLUTION OF THE POISSON EQUATION BY THE FINITE DIFFERENCE-CHEBYSHEV-TAU METHOD

A new finite difference-Chebyshev-Tau method for the solution of the two-dimensional Poisson equation is presented. Some of the numerical results are also presented which indicate that the method is satisfactory and compatible to other methods.