非线性发展型方程的Legendre拟谱逼近

本文考虑非稳态Burgers方程的拟谱逼近，构造了一类Legendre拟谱计算格式并证明了其收敛性，数值结果显示了格式的有效性。     

解高维广义BBM方程的谱方法和拟谱方法

在非线性色散介质的长波研究中,Benjanin,Bona和Mahony等人提出并讨论了BBM方程。这类方程在许多数学物理问题中出现,如热力学中的双温热传导问题、在岩石裂缝中的渗流问题等,因而引起了人们的重视。之后,Goldstein,Avrin,郭柏灵等进一步研究了高维广义BBM方程。这类方程的数值分析很多,但主要是差分法和有限元法,如[9-10],[11]在一维情形下用谱方法和拟谱方法作了研究。本文讨论高维     

热传导方程的一类半无界问题

本文推广了对称开拓法，解决了热传导方程的在有限端具有比较广泛的一类边值条件的半无界问题，并求出解的表示式。     

非线性Cahn-Hilliard方程的拟谱算法

本文对非线性Cahn-Hilliard方程构造了拟谱格式,证明了该格式的收敛性和稳定性,给出了数值例子.     

多维区域中非线性偏微分方程的修正Laguerre谱与拟谱方法

研究多维区域中非线性偏微分方程的谱与拟谱方法．建立了修正Laguerre正交逼近与插值结果,这些结果对于建立和分析无界区域中的数值方法起着重要的作用．作为结果的一个应用,研究了二维无界区域中的Logistic方程的修正Laguerre谱格式,证明了它的稳定性和收敛性．数值试验结果表明所提出方法具有很高的精度,与理论分析结果完全吻合．     

非线性热传导方程的几类精确解

通过引入恰当的试探函数,将非线性热传导方程化为易于求解的常微分方程组并对其求解,进而得到非线性热传导方程的孤波解、奇异行波解、三角函数周期波解等一些不同形式的行波解.     

Cahn-Hilliard方程的拟谱逼近

该文讨论用Ｌｅｇｅｎｄｒｅ拟谱方法数值求解非线性Ｃａｈｎ Ｈｉｌｌｉａｒｄ方程的Ｄｉｒｉｃｈｌｅｔ问题．建立了其半离散和全离散逼近格式，它们保持原问题能量耗散的性质．证明了离散解的存在唯一性，并给出了最佳误差估计．数值实验也证实了我们的结果．     

Laguerre-Galerkin method for nonlinear partial differential equations on a semi-infinite interval

Summary. A Laguerre-Galerkin method is proposed and analyzed for the Burgers equation and Benjamin-Bona-Mahony (BBM) equation on a semi-infinite interval. By reformulating these equations with suitable functional transforms, it is shown that the Laguerre-Galerkin approximations are convergent on a semi-infinite interval with spectral accuracy. An efficient and accurate algorithm based on the Laguerre-Galerkin approximations to the transformed equations is developed and implemented. Numerical results indicating the high accuracy and effectiveness of this algorithm are presented. Received October 6, 1997 / Revised version received July 22, 1999 / Published online June 21, 2000     

Modified Laguerre pseudospectral method refined by multidomain Legendre pseudospectral approximation

A modified Laguerre pseudospectral method is proposed for differential equations on the half-line. The numerical solutions are refined by multidomain Legendre pseudospectral approximation. Numerical results show the spectral accuracy of this approach. Some approximation results on the modified Laguerre and Legendre interpolations are established. The convergence of proposed method is proved.     

Stair Laguerre pseudospectral method for differential equations on the half line

A stair Laguerre pseudospectral method is proposed for numerical solutions of differential equations on the half line. Some approximation results are established. A stair Laguerre pseudospcetral scheme is constructed for a model problem. The convergence is proved. The numerical results show that this new method provides much more accurate numerical results than the standard Laguerre spectral method. Dedicated to Charles A. Micchelli on the occasion of his 60th birthday Mathematics subject classifications (2000) 65N35, 41A10. Li-lian Wang: The work of this author is partially supported by The Shanghai Natural Science Foundation N. 00JC14057, The Shanghai Natural Science Foundation for Youth N. 01QN85 and The Special Funds for Major Specialities of Shanghai Education Committee. Ben-yu Guo: The work of this author is partially supported by The Special Funds for Major State Basic Research Projects of China G1999032804, The Shanghai Natural Science Foundation N. 00JC14057 and The Special Funds for Major Specialities of Shanghai Education Committee.     

Fourier-Chebyshev pseudospectral method for two-dimensional vorticity equation

Summary A Fourier-Chebyshev pseudospectral scheme is proposed for two-dimensional unsteady vorticity equation. The generalized stability and convergence are proved strictly. The numerical results are presented.     

Blowup solutions in a problem for the nonlinear heat equation on a small interval

The article considers a one-dimensional quasi-linear heat equation with a volume heat source and a nonlinear thermal conductivity. The analysis is conducted for parameter values where selfsimilar solutions of the equations evolve in an LS-regime with blowup. Heat localization is observed in this case, and the combustion process in the developed stage is in the form of simple or complex structures of contracting half-width. We study the evolution dynamics of various initial distributions and their achievement of the self-similar regime, and also the dependence of the size of the localization region on the shape of the initial compactly supported distribution. The possibility of cyclic evolution of solutions against the background of overall growth with blowup is demonstrated. We particularly focus on the case when the size of the spatial region is much less than the characteristic size of the localization region, and heat flow is obstructed by the physical boundaries. In this case all initial perturbations achieve the self-similar regime, but the corresponding scenario has certain specific features. We present an example of formation of a complex spatial structure that evolves with blowup on a small interval. Translated from Prikladnaya Matematika i Informatika, No. 29, 2008, pp. 88–112.     

On some method for solving a nonlinear heat equation

Some exact solutions to a nonlinear heat equation are constructed. An initial-boundary value problem is examined for a nonlinear heat equation. To construct solutions, the problem for a partial differential equation of the second order is reduced to a similar problem for a first order partial differential equation.     

A pseudospectral method for nonlinear Duffing equation involving both integral and non‐integral forcing terms

The Legendre pseudospectral method is developed for the numerical solution of nonlinear Duffing equation involving both integral and non‐integral forcing terms. By using differentiation matrix, the problem is reduced to the solution of a system of algebraic equations. The method is general, easy to implement, and yields very accurate results. Numerical experiments are presented to demonstrate the accuracy and the efficiency of the proposed computational procedure. Copyright © 2014 John Wiley & Sons, Ltd.     

A Fourier pseudospectral method for a generalized improved Boussinesq equation

In this article, we propose a Fourier pseudospectral method for solving the generalized improved Boussinesq equation. We prove the convergence of the semi‐discrete scheme in the energy space. For various power nonlinearities, we consider three test problems concerning the propagation of a single solitary wave, the interaction of two solitary waves and a solution that blows up in finite time. We compare our numerical results with those given in the literature in terms of numerical accuracy. The numerical comparisons show that the Fourier pseudospectral method provides highly accurate results. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 995–1008, 2015     

Subcritical nonlinear heat equation

We study asymptotic behavior in time of small solutions to nonlinear heat equations in subcritical case. We find a new family of self-similar solutions which change a sign. We show that solutions are stable in the neighborhood of these self-similar solutions.     

Laguerre polynomial solution of the nonlinear Boltzmann equation for the hard-sphere model

We consider a spatially homogeneous and isotropic gas consisting of hard-sphere molecules. A vector representation of the scattering kernel is used to adapt the original Boltzmann equation to the idealized geometrical situation. By means of an expansion of the distribution function in terms of Laguerre polynomials this scalar Boltzmann equation is transformed to a set of moment equations. All algebraized collision integrals can be evaluated analytically. We discuss the truncation of the moment equations necessary for the practical application of this method. The eigenvalues of the linearized relaxation problem show a good convergence with respect to the truncation index.

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Zusammenfassung Wir betrachten ein räumlich homogenes Gas harter Kugeln mit isotroper Geschwindigkeitsverteilung. Mit Hilfe einer Vektordarstellung des Streukerns wird die nichtlineare Boltzmanngleichung den vereinfachten geometrischen Verhältnissen angepaßt. Die entstehende skalare kinetische Gleichung wird durch eine Laguerre-Reihenentwicklung der Teilchenverteilungsdichte in ein System von Momentegleichungen übergeführt. Sämtliche algebraisierten Stoßintegrale erweisen sich als analytisch lösbar. Wir diskutieren den für den praktischen Gebrauch der Methode notwendigen Abbruch des Systems der Momentegleichungen. Die Eigenwerte des linearisierten Relaxationsproblems zeigen eine rasche Konvergenz bezüglich einer Steigerung des Abbruchindex.