共查询到20条相似文献,搜索用时 10 毫秒
1.
L. M. Skvortsov 《Computational Mathematics and Mathematical Physics》2007,47(6):915-923
An explicit multistep method of variable order for integrating stiff systems with high accuracy and low computational costs is examined. To stabilize the computational scheme, componentwise estimates are used for the eigenvalues of the Jacobian matrix having the greatest moduli. These estimates are obtained at preliminary stages of the integration step. Examples are given to demonstrate that, for certain stiff problems, the method proposed is as efficient as the best implicit methods. 相似文献
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In this paper, we propose new technique for solving stiff system of ordinary differential equations. This algorithm is based on Laplace transform and homotopy perturbation methods. The new technique is applied to solving two mathematical models of stiff problem. We show that the present approach is relatively easy, efficient and highly accurate. 相似文献
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Franco Obersnel Pierpaolo Omari 《Proceedings of the American Mathematical Society》2007,135(7):2055-2058
We extend a result of J. Andres and K. Pastor concerning scalar time-periodic first order ordinary differential equations without uniqueness, by proving that the existence of just one subharmonic implies the existence of large sets of subharmonics of all given orders. Since these periodic solutions must coexist with complicated dynamics, we might paraphrase T. Y. Li and J. A. Yorke by loosely saying that in this setting even period two implies chaos. Similar results are obtained for a class of differential inclusions.
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Shoufu LI 《Frontiers of Mathematics in China》2009,4(1):23-48
In this review, we present the recent work of the author in comparison with various related results obtained by other authors
in literature. We first recall the stability, contractivity and asymptotic stability results of the true solution to nonlinear
stiff Volterra functional differential equations (VFDEs), then a series of stability, contractivity, asymptotic stability
and B-convergence results of Runge-Kutta methods for VFDEs is presented in detail. This work provides a unified theoretical foundation
for the theoretical and numerical analysis of nonlinear stiff problems in delay differential equations (DDEs), integro-differential
equations (IDEs), delayintegro-differential equations (DIDEs) and VFDEs of other type which appear in practice.
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6.
线性微分方程的微分算子级数解法 总被引:15,自引:0,他引:15
介绍了微分算子级数法及其求解线性常微分方程通解、特解的原理、方法和实例.这个方法和其它解法的差别,在于不借助其它学科知识的启示,直接通过方程中微分算子的运算求出方程的特解或通解. 相似文献
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Higinio Ramos Jesús Vigo‐Aguiar 《Numerical Methods for Partial Differential Equations》2007,23(5):1110-1121
A new BDF‐type scheme is proposed for the numerical integration of the system of ordinary differential equations that arises in the Method of Lines solution of time‐dependent partial differential equations. This system is usually stiff, so it is desirable for the numerical method to solve it to have good properties concerning stability. The method proposed in this article is almost L‐stable and of algebraic order three. Numerical experiments illustrate the performance of the new method on different stiff systems of ODEs after discretizing in the space variable some PDE problems. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007 相似文献
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本讨论二阶微分方程的第二边值问题具有模糊不确定性时,运用模糊仿真原理和差分方法,求其边值问题的数值解法。 相似文献
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In this paper we analyze the problem of adaptivity for one-step numerical methods for solving ODEs, both IVPs and BVPs, with a view to generating grids of minimal computational cost for which the local error is below a prescribed tolerance (optimal grids). The grids are generated by introducing an auxiliary independent variable τ and finding a grid deformation map, t=Θ(τ), that maps an equidistant grid {τj} to a non-equidistant grid in the original independent variable, {tj}. An optimal deformation map Θ is determined by a variational approach. Finally, we investigate the cost of the solution procedure and compare it to the cost of using equidistant grids. We show that if the principal error function is non-constant, an adaptive method is always more efficient than a non-adaptive method. 相似文献
10.
Jiří Taufer 《Applications of Mathematics》2004,49(2):141-164
The paper is devoted to solving boundary value problems for self-adjoint linear differential equations of 2nth order in the case that the corresponding differential operator is self-adjoint and positive semidefinite. The method proposed consists in transforming the original problem to solving several initial value problems for certain systems of first order ODEs. Even if this approach may be used for quite general linear boundary value problems, the new algorithms described here exploit the special properties of the boundary value problems treated in the paper. As a consequence, we obtain algorithms that are much more effective than similar ones used in the general case. Moreover, it is shown that the algorithms studied here are numerically stable. 相似文献
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R. S. Temsah 《Numerical Algorithms》2008,47(1):63-80
Spectral methods with interface point are presented to deal with some singularly perturbed third order boundary value problems
of reaction-diffusion and convection-diffusion types. First, linear equations are considered and then non-linear equations.
To solve non-linear equations, Newton’s method of quasi-linearization is applied. The problem is reduced to two systems of
ordinary differential equations. And, then, each system is solved using spectral collocation methods. Our numerical experiments
show that the proposed methods are produce highly accurate solutions in little computer time when compared with the other
methods available in the literature.
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Theodore S. Papatheodorou Anastasia N. Kandili 《Journal of Computational and Applied Mathematics》2009
The unified transform method of A. S. Fokas has led to important new developments, regarding the analysis and solution of various types of linear and nonlinear PDE problems. In this work we use these developments and obtain the solution of time-dependent problems in a straightforward manner and with such high accuracy that cannot be reached within reasonable time by use of the existing numerical methods. More specifically, an integral representation of the solution is obtained by use of the A. S. Fokas approach, which provides the value of the solution at any point, without requiring the solution of linear systems or any other calculation at intermediate time levels and without raising any stability problems. For instance, the solution of the initial boundary value problem with the non-homogeneous heat equation is obtained with accuracy 10−15, while the well-established Crank–Nicholson scheme requires 2048 time steps in order to reach a 10−8 accuracy. 相似文献
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A. F. Latypov Yu. V. Nikulichev 《Computational Mathematics and Mathematical Physics》2007,47(2):227-237
Families of A-, L-, and L(δ)-stable methods are constructed for solving the Cauchy problem for a system of ordinary differential equations (ODEs). The L(δ)-stability of a method with a parameter δ ∈ (0, 1) is defined. The methods are based on the representation of the right-hand sides of an ODE system at the step h in terms of two-or three-point Hermite interpolating polynomials. Comparative results are reported for some test problems. The multipoint Hermite interpolating polynomials are used to derive formulas for evaluating definite integrals. Error estimates are given. 相似文献
16.
We employ B-series to analyze the order of Runge-Kutta (RK) methods that use simple iteration, modified Newton iteration or full Newton iteration to compute their internal stage values. Our assumptions about the initial guess for the internal stage values cover a wide range of practical schemes. Moreover, the analytical techniques developed in this paper can be applied in many other similar contexts.This work was supported in part by the Natural Sciences and Engineering Research Council of Canada, the Information Technology Research Centre of Ontario, and the Norwegian Research Council for Science and Humanities. 相似文献
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1.IntroductionInthedevelopmentofnewelectricalcircuits,thesimulationofthebehaviourofthecircuithasbecomeanessentialtoolforelectricalengineers.Fromthelayoutofthecircuitanonlinearsystemofordinarydifferentialequationsisgeneratedwhichdescribesthedynamicalbehaviourofthecircuit.Inthesimulationofverylargescaleintegrated(VLSI)circuitsthedimensionofthesystemofODEscanbecomeverylarge.Moreoversincethesystemisstiff,solvingthesesystemsisaverycomputionallyintensivetaskandtheuseofsupercomputersbecomesin-evit… 相似文献
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本文利用李雅普诺夫第二方法[1]给出了至少有一个特征根具有正实部的四阶变系数线性微分方程解的不稳定性的充分条件。 相似文献
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变系数线性微分方程初值问题数值解的小波方法 总被引:1,自引:0,他引:1
通过利用小波尺度函数的正交性并结合配点法 ,本文给出了一种求解变系数线性微分方程初值问题数值解的小波算法 .在一定的假设条件下 ,对算法的收敛性进行了理论分析 .最后 ,我们还给出了一个具体的数值计算例子 . 相似文献