UNIFORM ANALYTIC CONSTRUCTION OF WAVELET ANALYSIS FILTERS BASED ON SINE AND COSINE TRIGONOMETRIC FUNCTIONS

Based on sine and cosine functions, the compactly supported orthogonal wavelet filter coefficients with arbitrary length are constructed for the first time. When N = 2(k-1) and N = 2k, the unified analytic constructions of orthogonal wavelet filters are put forward, respectively. The famous Daubechies filter and some other well-known wavelet filters are tested by the proposed novel method which is very useful for wavelet theory research and many application areas such as pattern recognition.

A constructive method for approximating trigonometric functions and their integrals

This paper presents an interpolation-based method(IBM)for approximating some trigonometric functions or their integrals as well.It provides two-sided bounds for each function,which also achieves much better approximation effects than those of prevailing methods.In principle,the IBM can be applied for bounding more bounded smooth functions and their integrals as well,and its applications include approximating the integral of sin(x)/x function and improving the famous square root inequalities.

Fractional Weierstrass Function by Application of Jumarie Fractional Trigonometric Functions and Its Analysis

The classical example of no-where differentiable but everywhere continuous function is Weierstrass function. In this paper we have defined fractional order Weierstrass function in terms of Jumarie fractional trigonometric functions. The H?lder exponent and Box dimension of this new function have been evaluated here. It has been established that the values of H?lder exponent and Box dimension of this fractional order Weierstrass function are the same as in the original Weierstrass function. This new development in generalizing the classical Weierstrass function by use of fractional trigonometric function analysis and fractional derivative of fractional Weierstrass function by Jumarie fractional derivative, establishes that roughness indices are invariant to this generalization.

The Maximum Trigonometric Degrees of Quadrature Formulae with Prescribed Nodes

The purpose of this paper is to study the maximum trigonometric degree of the quadrature formula associated with m prescribed nodes and n unknown additional nodes in the interval（-π, π]. We show that for a fixed n, the quadrature formulae with m and m ＋ 1 prescribed nodes share the same maximum degree if m is odd. We also give necessary and sufficient conditions for all the additional nodes to be real, pairwise distinct and in the interval（-π, π] for even m, which can be obtained constructively. Some numerical examples are given by choosing the prescribed nodes to be the zeros of Chebyshev polynomials of the second kind or randomly for m ≥ 3.

A New Method for Calculating Inverse Involute Functions with High Precision

Two equations are presented for calculating inverse involute functions. Precise values can be obtained easily with an ordinary calculator.

ON GENERALIZED COMPLETE(p,q)-ELLIPTIC INTEGRALS

In this paper,we study the generalized complete(p,q)-elliptic integrals of the first and second kind as an application of generalized trigonometric functions with two parameters,and establish the monotonicity,generalized convexity and concavity of these functions.In particular,some Turán type inequalities are given.Finally,we also show some new series representations of these functions by applying Alzer and Richard's methods.

A Generalized F-expansion Method and Its Application in High-Dimensional Nonlinear Evolution Equation

A generalized F-expansion method is introduced and applied to (3+ 1)-dimensional Kadomstev-Petviashvili(KP) equation. As a result, some new Jacobi elliptic function solutions of the equation are found, from which the trigonometric function solutions and the solitary wave solutions can be obtained. The method can also be extended to other types of nonlinear evolution equations in mathematical physics.

New Exact Solutions of the Integrable Broer-Kaup Equations in （2＋ 1）-DimensionalSpaces

In this paper,by improving some procedure of extended tanh-function method,some new exact solutions to the integrable Broer-Kaup equations in(2 + 1)-dimensional spaces are obtained,which include soliton-like solutions,solitary wave solutions,trigonometric function solutions,and rational solutions.

A Differential Quadrature Based Approach for Volterra Partial Integro-Differential Equation with a Weakly Singular Kernel

Differential quadrature method is employed by numerous researchers due to its numerical accuracy and computational efficiency,and is mentioned as potential alternative of conventional numerical methods.In this paper,a differential quadrature based numerical scheme is developed for solving volterra partial integro-differential equation of second order having a weakly singular kernel.The scheme uses cubic trigonometric B-spline functions to determine the weighting coefficients in the differential quadrature approximation of the second order spatial derivative.The advantage of this approximation is that it reduces the problem to a first order time dependent integro-differential equation(IDE).The proposed scheme is obtained in the form of an algebraic system by reducing the time dependent IDE through unconditionally stable Euler backward method as time integrator.The scheme is validated using a homogeneous and two nonhomogeneous test problems.Conditioning of the system matrix and numerical convergence of the method are analyzed for spatial and temporal domain discretization parameters.Comparison of results of the present approach with Sinc collocation method and quasi-wavelet method are also made.

Jacobian Elliptic Function Method and Solitary Wave Solutions for Hybrid Lattice Equation

In this paper, we have successfully extended the Jacobian elliptic function expansion approach to nonlinear differential-difference equations. The Hybrid lattice equation is chosen to illustrate this approach. As a consequence, twelve families of Jacobian elliptic function solutions with different parameters of the Hybrid lattice equation are obtained. When the modulus m → 1 or O, doubly-periodic solutions degenerate to solitonic solutions and trigonometric function solutions, respectively.

Traveling Wave Solutions for Generalized Bretherton Equation

This paper studies the Generalized Bretherton equation using trigonometric function method including the sech-function method, the sine-cosine function method, and the tanh-function method, and He＇s semi-inverse method （He＇s variational method）. Various traveling wave solutions are obtained, revealing an intrinsic relationship among the amplitude, frequency, and wave speed.

Variable-Coefficient Mapping Method Based on Elliptical Equation and Exact Solutions to Nonlinear SchrSdinger Equations with Variable Coefficient

In this paper, by means of the variable-coefficient mapping method based on elliptical equation, we obtain explicit solutions of nonlinear Schrodinger equation with variable-coefficient. These solutions include Jacobian elliptic function solutions, solitary wave solutions, soliton-like solutions, and trigonometric function solutions, among which some are found for the first time. Six figures are given to illustrate some features of these solutions. The method can be applied to other nonlinear evolution equations in mathematical physics.

Karhunen-loéve expansion for random earthquake excitations

This paper develops a trigonometric-basis-fimction based Karhunen-Loeve （KL） expansion for simulating random earthquake excitations with known covariance functions. The methods for determining the number of the KL terms and defining the involved random variables are described in detail. The simplified form of the KL expansion is given, whereby the relationship between the KL expansion and the spectral representation method is investigated and revealed. The KL expansion is of high efficiency for simulating long-term earthquake excitations in the sense that it needs a minimum number of random variables, as compared with the spectral representation method. Numerical examples demonstrate the convergence and accuracy of the KL expansion for simulating two commonly-used random earthquake excitation models and estimating linear and nonlinear random responses to the random excitations.

Exact Solutions of (2+1)-Dimensional Boiti-Leon-Pempinelle Equation with (G'/G)-Expansion Method

In this paper,we construct exact solutions for the (2+1)-dimensional Boiti-Leon-Pempinelle equation byusing the (G'/G)-expansion method,and with the help of Maple.As a result,non-travelling wave solutions with threearbitrary functions are obtained including hyperbolic function solutions,trigonometric function solutions,and rationalsolutions.This method can be applied to other higher-dimensional nonlinear partial differential equations.

New exact solutions of nonlinear differential-difference equations with symbolic computation

In this paper, the Toda equation and the discrete nonlinear Schrdinger equation with a saturable nonlinearity via the discrete ＂ （G′/G＂）-expansion method are researched. As a result, with the aid of the symbolic computation, new hyperbolic function solution and trigonometric function solution with parameters of the Toda equation are obtained. At the same time, new envelop hyperbolic function solution and envelop trigonometric function solution with parameters of the discrete nonlinear Schro¨dinger equation with a saturable nonlinearity are obtained. This method can be applied to other nonlinear differential-difference equations in mathematical physics.

New Exact Solutions to （2＋1）-Dimensional Variable Coefficients Broer-Kaup Equations

In this paper, using the variable coefficient generalized projected Rieatti equation expansion method, we present explicit solutions of the （2＋1）-dimensional variable coefficients Broer-Kaup （VCBK） equations. These solutions include Weierstrass function solution, solitary wave solutions, soliton-like solutions and trigonometric function solutions. Among these solutions, some are found for the first time. Because of the three or four arbitrary functions, rich localized excitations can be found.

A Comparative Numerical Study of Parabolic Partial Integro-Differential Equation Arising from Convection-Diffusion

This article studies the development of two numerical techniques for solving convection-diffusion type partial integro-differential equation(PIDE)with a weakly singular kernel.Cubic trigonometric B-spline(CTBS)functions are used for interpolation in both methods.The first method is CTBS based collocation method which reduces the PIDE to an algebraic tridiagonal system of linear equations.The other method is CTBS based differential quadrature method which converts the PIDE to a system of ODEs by computing spatial derivatives as weighted sum of function values.An efficient tridiagonal solver is used for the solution of the linear system obtained in the first method as well as for determination of weighting coefficients in the second method.An explicit scheme is employed as time integrator to solve the system of ODEs obtained in the second method.The methods are tested with three nonhomogeneous problems for their validation.Stability,computational efficiency and numerical convergence of the methods are analyzed.Comparison of errors in approximations produced by the present methods versus different values of discretization parameters and convection-diffusion coefficients are made.Convection and diffusion dominant cases are discussed in terms of Peclet number.The results are also compared with cubic B-spline collocation method.

Best m-term One-sided Trigonometric Approximation of Some Function Classes Defined by a Kind of Multipliers

In this paper, we continue studying the so-called non-linear best m-term one-sided approximation problems and obtain the asymptotic estimations of non-linear best m-term one-sided trigonometric approximation under the norm Lp （1 ≤ p ≤ ∞） of multiplier function classes and the corresponding m-term Greedy-liked one-sided trigonometric approximation results.

Application of the Residue Theorem to Trigonometric Sum Identities

By evaluating a contour integral with the Cauchy residue theorem, we prove a general summation formula on trigonometric sum, which contains several interesting trigonometric identities as special cases.

Path-tracking control based on a dynamic trigonometric function

With the rapid development of the modern vehicle industry,the automated control of new vehicles is in increasing demand.However,traditional course control has been unable to meet the actual needs of such demand.To solve this problem,more precise pathtracking control technologies have attracted increased attention.This paper presents a new algorithm based on the latitude and longitude information,as well as a dynamic trigonometric function,to improve the accuracy of position deviation.First,the algorithm takes the course deviation and adjustment time as the optimization objectives and the given path and speed as the constraints.The controller continuously adjusts the output through a cyclic“adjustment and detection”process.Second,through an integration of the steering,positioning,and speed control systems,an experimental platform of a path-tracking control system based on the National Instruments(NI)myRIO controller and LabVIEW was developed.In addition,path-tracking experiments were carried out along a linear path,while changing lanes,and on a curved path.When comparing and analyzing the experimental results,it can be seen that the average deviation in lateral displacement along the linear and curved paths was 0.32 and
0.8 cm,and the standard deviation of the lateral displacement was 2.65 and 2.39 cm,respectively.When changing lanes,the total adjustment time for the vehicle close to the target line to reach stability was about 1.5 s.Finally,the experimental results indicate that the new algorithm achieves good stability and high control accuracy,and can overcome directional and positional errors caused by road interference while driving,meeting the precision requirements of automated vehicle control.