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 constructio...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.展开更多
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 approximat...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.展开更多
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 trigonome...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.展开更多
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 sol...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 and0.8 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.展开更多
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 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.展开更多
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...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.展开更多
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 ...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.展开更多
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,generali...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.展开更多
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 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.展开更多
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 t...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.展开更多
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)functio...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.展开更多
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.
This article is devoted to a newly introduced numerical method for time-fractional dispersive partial differential equation in a multidimensional space.The time-fractional dispersive partial differential equation play...This article is devoted to a newly introduced numerical method for time-fractional dispersive partial differential equation in a multidimensional space.The time-fractional dispersive partial differential equation plays a great role in solving the problems arising in ocean science and engineering.The numerical technique comprises of Sumudu transform,homotopy perturbation scheme and He’s polynomial,namely homotopy perturbation Sumudu transform method(HPSTM)is efficiently used to examine time-fractional dispersive partial differential equation of third order in multi-dimensional space.The approximate analytic solution of the time-fractional dispersive partial differential equation of third-order in multi-dimensional space obtained by HPSTM is compared with exact solution as well as the solution obtained by using Adomain decomposition method.The results derived with the aid of two techniques are in a good agreement and consequently these techniques may be considered as an alternative and efficient approach for solving fractional partial differential equations.Several test problems are experimented to confirm the accuracy and efficiency of the proposed methods.展开更多
The mechanical behavior of advanced composites can be modeled mathematically through unknown variables and Shear Strain Thickness Functions(SSTFs). Such SSTFs can be of polynomial or non-polynomial nature and some par...The mechanical behavior of advanced composites can be modeled mathematically through unknown variables and Shear Strain Thickness Functions(SSTFs). Such SSTFs can be of polynomial or non-polynomial nature and some parameters of non-polynomial SSTFs can be optimized to get optimal results. In this paper, these parameters are called ‘‘r" and ‘‘s" and they are the argument of the trigonometric SSTFs introduced within the Carrera Unified Formulation(CUF). The Equivalent Single Layer(ESL) governing equations are obtained by employing the Principle of Virtual Displacement(PVD) and are solved using Navier method solution. Furthermore, trigonometric expansion with Murakami theory was implemented in order to reproduce the Zig-Zag effects which are important for multilayer structures. Several combinations of optimization parameters are evaluated and selected by different criteria of average error. Results of the present unified trigonometrical theory with CUF bases confirm that it is possible to improve the stress and displacement results through the thickness distribution of models with reduced unknown variables. Since the idea is to find a theory with reduced numbers of unknowns, the present method appears to be an appropriate technique to select a simple model. However these optimization parameters depend on the plate geometry and the order of expansion or unknown variables. So, the topic deserves further research.展开更多
文摘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.
基金Supported by the National Natural Science Foundation of China(61672009,61502130).
文摘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.
基金Board of Research in Nuclear Science (BRNS), Department of Atomic Energy Government of India
文摘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.
文摘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 and0.8 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.
基金The NSF (61033012,10801023,10911140268 and 10771028) of China
文摘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.
基金Supported by National Natural Science Foundation of China (Grant No. 10771016) supported by Shandong Agricultural University Youth Foundation
文摘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.
基金Project supported by the National Natural Science Foundation of China (Grant Nos.61072147,11071159)the Natural Science Foundation of Shanghai Municipality (Grant No.09ZR1410800)+1 种基金the Science Foundation of Key Laboratory of Mathematics Mechanization (Grant No.KLMM0806)the Shanghai Leading Academic Discipline Project (Grant Nos.J50101, S30104)
文摘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.
基金supported by the Natural Science Foundation of Shandong Province (ZR2019QA003 and ZR2018MF023)by the National Natural Science Foundation of China (11601036)by the Major Project of Binzhou University (2020ZD02)
文摘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.
文摘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.
文摘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.
文摘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.
文摘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.
文摘This article is devoted to a newly introduced numerical method for time-fractional dispersive partial differential equation in a multidimensional space.The time-fractional dispersive partial differential equation plays a great role in solving the problems arising in ocean science and engineering.The numerical technique comprises of Sumudu transform,homotopy perturbation scheme and He’s polynomial,namely homotopy perturbation Sumudu transform method(HPSTM)is efficiently used to examine time-fractional dispersive partial differential equation of third order in multi-dimensional space.The approximate analytic solution of the time-fractional dispersive partial differential equation of third-order in multi-dimensional space obtained by HPSTM is compared with exact solution as well as the solution obtained by using Adomain decomposition method.The results derived with the aid of two techniques are in a good agreement and consequently these techniques may be considered as an alternative and efficient approach for solving fractional partial differential equations.Several test problems are experimented to confirm the accuracy and efficiency of the proposed methods.
基金“Diseno y optimización de dispositivos de drenaje para pacientes con glaucoma mediante el uso de modelos computacionales de ojos"founded by Cienciactiva,CONCYTEC,under the contract number N°008-2016-FONDECYTfinancial support from the Peruvian Government
文摘The mechanical behavior of advanced composites can be modeled mathematically through unknown variables and Shear Strain Thickness Functions(SSTFs). Such SSTFs can be of polynomial or non-polynomial nature and some parameters of non-polynomial SSTFs can be optimized to get optimal results. In this paper, these parameters are called ‘‘r" and ‘‘s" and they are the argument of the trigonometric SSTFs introduced within the Carrera Unified Formulation(CUF). The Equivalent Single Layer(ESL) governing equations are obtained by employing the Principle of Virtual Displacement(PVD) and are solved using Navier method solution. Furthermore, trigonometric expansion with Murakami theory was implemented in order to reproduce the Zig-Zag effects which are important for multilayer structures. Several combinations of optimization parameters are evaluated and selected by different criteria of average error. Results of the present unified trigonometrical theory with CUF bases confirm that it is possible to improve the stress and displacement results through the thickness distribution of models with reduced unknown variables. Since the idea is to find a theory with reduced numbers of unknowns, the present method appears to be an appropriate technique to select a simple model. However these optimization parameters depend on the plate geometry and the order of expansion or unknown variables. So, the topic deserves further research.