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 t...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.展开更多
In this paper, we suggest a method for solving Fredholm integral equation of the first kind based on wavelet basis. The continuous Legendre and Chebyshev wavelets of the first, second, third and fourth kind on [0,1] a...In this paper, we suggest a method for solving Fredholm integral equation of the first kind based on wavelet basis. The continuous Legendre and Chebyshev wavelets of the first, second, third and fourth kind on [0,1] are used and are utilized as a basis in Galerkin method to approximate the solution of integral equations. Then, in some examples the mentioned wavelets are compared with each other.展开更多
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 homoge...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.展开更多
The second kind of modified Bessel function of order zero is the solutions of many problems in engineering. Modified Bessel equation is transformed by exponential transformation and expanded by J.P.Boyd's rational...The second kind of modified Bessel function of order zero is the solutions of many problems in engineering. Modified Bessel equation is transformed by exponential transformation and expanded by J.P.Boyd's rational Chebyshev basis.展开更多
In this paper, the Chebyshev wavelet method, constructed from the Chebyshev polynomial of the first kind is proposed to numerically simulate the single-phase flow of fluid in a reservoir. The method was used together ...In this paper, the Chebyshev wavelet method, constructed from the Chebyshev polynomial of the first kind is proposed to numerically simulate the single-phase flow of fluid in a reservoir. The method was used together with the operational matrices of integration which resulted in an algebraic system of equations. The system of equation was solved for the wavelet coefficient and used to construct the solutions. The efficiency and accuracy of the method were demonstrated through error measurements. Both the root mean square and the maximum absolute error analysis used in the study were within significantly close range. The Chebyshev wavelet collocation method subsequently was observed to closely approximate the analytic solution to the single phase flow model quite well.展开更多
The Fourier series of the 2π-periodic functions tg(x2)and 1sin(x)and some of their relatives (first of their integrals) are investigated and illustrated with respect to their convergence. These functions are Generali...The Fourier series of the 2π-periodic functions tg(x2)and 1sin(x)and some of their relatives (first of their integrals) are investigated and illustrated with respect to their convergence. These functions are Generalized functions and the convergence is weak convergence in the sense of the convergence of continuous linear functionals defining them. The figures show that the approximations of the Fourier series possess oscillations around the function which they represent in a broad band embedding them. This is some analogue to the Gibbs phenomenon. A modification of Fourier series by expansion in powers cosn(x)for the symmetric part of functions and sin(x)cosn−1(x)for the antisymmetric part (analogous to Taylor series) is discussed and illustrated by examples. The Fourier series and their convergence behavior are illustrated also for some 2π-periodic delta-function-like sequences connected with the Poisson theorem showing non-vanishing oscillations around the singularities similar to the Gibbs phenomenon in the neighborhood of discontinuities of functions. .展开更多
The basic aim of this paper is to introduce and describe an efficient numerical scheme based on spectral approach coupled with Chebyshev wavelets for the approximate solutions of Klein-Gordon and Sine-Gordon equations...The basic aim of this paper is to introduce and describe an efficient numerical scheme based on spectral approach coupled with Chebyshev wavelets for the approximate solutions of Klein-Gordon and Sine-Gordon equations. The main characteristic is that, it converts the given problem into a system of algebraic equations that can be solved easily with any of the usual methods. To show the accuracy and the efficiency of the method, several benchmark problems are implemented and the comparisons are given with other methods existing in the recent literature. The results of numerical tests confirm that the proposed method is superior to other existing ones and is highly accurate展开更多
The wavelet approach is introduced to study the influence of the natural convection stagnation point flow of the Williamson fluid in the presence of thermophysical and Brownian motion effects. The thermal radiation ef...The wavelet approach is introduced to study the influence of the natural convection stagnation point flow of the Williamson fluid in the presence of thermophysical and Brownian motion effects. The thermal radiation effects are considered along a permeable stretching surface. The nonlinear problem is simulated numerically by using a novel algorithm based upon the Chebyshev wavelets. It is noticed that the velocity of the Williamson fluid increases for assisting flow cases while decreases for opposing flow cases when the unsteadiness and suction parameters increase, and the magnetic effect on the velocity increases for opposing flow cases while decreases for assisting flow cases. When the thermal radiation parameter, the Dufour number, and Williamson’s fluid parameter increase, the temperature increases for both assisting and opposing flow cases. Meanwhile, the temperature decreases when the Prandtl number increases. The concentration decreases when the Soret parameter increases, while increases when the Schmidt number increases. It is perceived that the assisting force decreases more than the opposing force. The findings endorse the credibility of the proposed algorithm, and could be extended to other nonlinear problems with complex nature.展开更多
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 ...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.展开更多
文摘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.
文摘In this paper, we suggest a method for solving Fredholm integral equation of the first kind based on wavelet basis. The continuous Legendre and Chebyshev wavelets of the first, second, third and fourth kind on [0,1] are used and are utilized as a basis in Galerkin method to approximate the solution of integral equations. Then, in some examples the mentioned wavelets are compared with each other.
基金Hunan Provincial Natural Science Foundation Under Grant No.02JJY2085
文摘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.
文摘The second kind of modified Bessel function of order zero is the solutions of many problems in engineering. Modified Bessel equation is transformed by exponential transformation and expanded by J.P.Boyd's rational Chebyshev basis.
文摘In this paper, the Chebyshev wavelet method, constructed from the Chebyshev polynomial of the first kind is proposed to numerically simulate the single-phase flow of fluid in a reservoir. The method was used together with the operational matrices of integration which resulted in an algebraic system of equations. The system of equation was solved for the wavelet coefficient and used to construct the solutions. The efficiency and accuracy of the method were demonstrated through error measurements. Both the root mean square and the maximum absolute error analysis used in the study were within significantly close range. The Chebyshev wavelet collocation method subsequently was observed to closely approximate the analytic solution to the single phase flow model quite well.
文摘The Fourier series of the 2π-periodic functions tg(x2)and 1sin(x)and some of their relatives (first of their integrals) are investigated and illustrated with respect to their convergence. These functions are Generalized functions and the convergence is weak convergence in the sense of the convergence of continuous linear functionals defining them. The figures show that the approximations of the Fourier series possess oscillations around the function which they represent in a broad band embedding them. This is some analogue to the Gibbs phenomenon. A modification of Fourier series by expansion in powers cosn(x)for the symmetric part of functions and sin(x)cosn−1(x)for the antisymmetric part (analogous to Taylor series) is discussed and illustrated by examples. The Fourier series and their convergence behavior are illustrated also for some 2π-periodic delta-function-like sequences connected with the Poisson theorem showing non-vanishing oscillations around the singularities similar to the Gibbs phenomenon in the neighborhood of discontinuities of functions. .
文摘The basic aim of this paper is to introduce and describe an efficient numerical scheme based on spectral approach coupled with Chebyshev wavelets for the approximate solutions of Klein-Gordon and Sine-Gordon equations. The main characteristic is that, it converts the given problem into a system of algebraic equations that can be solved easily with any of the usual methods. To show the accuracy and the efficiency of the method, several benchmark problems are implemented and the comparisons are given with other methods existing in the recent literature. The results of numerical tests confirm that the proposed method is superior to other existing ones and is highly accurate
基金Project supported by the National Natural Science Foundation of China(Nos.51709191,51706149,and 51606130)the Key Laboratory of Advanced Reactor Engineering and Safety,Ministry of Education of China(No.ARES-2018-10)the State Key Laboratory of Hydraulics and Mountain River Engineering of Sichuan University of China(No.Skhl1803)
文摘The wavelet approach is introduced to study the influence of the natural convection stagnation point flow of the Williamson fluid in the presence of thermophysical and Brownian motion effects. The thermal radiation effects are considered along a permeable stretching surface. The nonlinear problem is simulated numerically by using a novel algorithm based upon the Chebyshev wavelets. It is noticed that the velocity of the Williamson fluid increases for assisting flow cases while decreases for opposing flow cases when the unsteadiness and suction parameters increase, and the magnetic effect on the velocity increases for opposing flow cases while decreases for assisting flow cases. When the thermal radiation parameter, the Dufour number, and Williamson’s fluid parameter increase, the temperature increases for both assisting and opposing flow cases. Meanwhile, the temperature decreases when the Prandtl number increases. The concentration decreases when the Soret parameter increases, while increases when the Schmidt number increases. It is perceived that the assisting force decreases more than the opposing force. The findings endorse the credibility of the proposed algorithm, and could be extended to other nonlinear problems with complex nature.
文摘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.
