共查询到19条相似文献,搜索用时 140 毫秒
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应用Gauss全局径向基函数来模拟波浪浅水变形波高变化方程中的未知函数,经实例分析探讨得到了一种可用于求解该方程数值解的新方法,并将其计算结果与常用数值分析方法得到的数值解相互对比印证,证明了基于Gauss全局径向基函数法计算结果的正确性.经验证,Gauss径向基函数法的平均计算误差相比其他方法均要小,表明该方法拥有更高的计算精度.同时,根据Gauss全局径向基函数的逼近结果,得出了浅水变形波高变化微分方程数值解的拟合函数,在实际工程中,可以利用该拟合函数来代替原方程的解析解,研究成果可为求解近岸浅水区域波浪运动提供一种新思路. 相似文献
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针对非线性Black-Scholes方程,基于quasi-Shannon小波函数给出了一种求解非线性偏微分方程的自适应多尺度小波精细积分法.该方法首先利用插值小波理论构造了用于逼近连续函数的多尺度小波插值算子,利用该算子可以将非线性Black-Scholes方程自适应离散为非线性常微分方程组;然后将用于求解常微分方程组的精细积分法和小波变换的动态过程相结合,并利用非线性处理技术(如同伦分析技术)可有效求解非线性Black-Scholes方程.数值结果表明了该方法在数值精度和计算效率方面的优越性. 相似文献
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采用单位分解径向基函数(radial basis function partition of unity,RBF-PU)方法,数值求解了二维非局部扩散问题和近场动力学问题。主要思想是对求解区域进行局部划分,在局部子区域上分别进行函数逼近,然后加权得到未知函数的全局逼近。这种基于方程强形式的径向基函数方法在求解非局部问题时,不需要处理网格与球形邻域求交的问题,避免了额外的一层积分计算,实施简便,计算量小。数值实验显示计算结果与解析解吻合较好,RBF-PU方法可以准确有效地求解非局部扩散方程和近场动力学方程。 相似文献
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基于弹性力学边界元方法理论,将边界元法与双互易法结合,采用指数型基函数对非齐次项进行插值得到双互易边界积分方程.将边界积分方程离散为代数方程组,利用已知边界条件和方程特解求解方程组,得出域内位移和边界面力.指数型基函数的形状参数是由插值点最近距离的最小值决定,采用这种形状参数变化方案,分析径向基函数(RBF)插值精度以及插值稳定性.再次将指数型基函数应用到双互易边界元法中,分析双互易边界元方法下计算精度及稳定性,验证了指数型插值函数作为双互易边界元方法的径向基函数解决弹性力学域内体力项问题的有效性. 相似文献
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《Numerical Methods for Partial Differential Equations》2018,34(3):959-981
In this article, integrated radial basis functions (IRBFs) are used for Hermite interpolation in the solution of differential equations, resulting in a new meshless symmetric RBF method. Both global and local approximation‐based schemes are derived. For the latter, the focus is on the construction of compact approximation stencils, where a sparse system matrix and a high‐order accuracy can be achieved together. Cartesian‐grid‐based stencils are possible for problems defined on nonrectangular domains. Furthermore, the effects of the RBF width on the solution accuracy for a given grid size are fully explored with a reasonable computational cost. The proposed schemes are numerically verified in some elliptic boundary‐value problems governed by the Poisson and convection‐diffusion equations. High levels of the solution accuracy are obtained using relatively coarse discretisations. 相似文献
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A numerical method for solving the nonlinear Fredholom integral equations is presented. The method is based on interpolation by radial basis functions (RBF) to approximate the solution of the Fredholm nonlinear integral equations. Several examples are given and numerical examples are presented to demonstrate the validity and applicability of the method. 相似文献
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The numerical differentiation is often used when dealing with the differential equations. Using the numerical differentiation, the differential equations can be transformed into algebraic equations. Then we can get the numerical solution from the algebraic equations. But the numerical differentiation process is very sensitive to even a small level of errors. In contrast, it is expected that on average the numerical integration process is much less sensitive to errors. In this paper, we provide a new method using the DQ method based on the interpolation of the highest derivative (DQIHD) for the differential equations. The original function is then obtained by integration. In this paper, the DQIHD method was applied to the buckling analysis of thin isotropic plates and Winkler plates, the numerical results agree well with the analytic solutions, and the results show that our method is of high accuracy, of good convergence with little computational efforts. And it is easy to deal with the boundary conditions. 相似文献
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人工神经网络近年来得到了快速发展,将此方法应用于数值求解偏微分方程是学者们关注的热点问题.相比于传统方法其具有应用范围广泛(即同一种模型可用于求解多种类型方程)、网格剖分条件要求低等优势,并且能够利用训练好的模型直接计算区域中任意点的数值.该文基于卷积神经网络模型,对传统有限体积法格式中的权重系数进行优化,以得到在粗粒度网格下具有较高精度的新数值格式,从而更适用于复杂问题的求解.该网络模型可以准确、有效地求解Burgers方程和level set方程,数值结果稳定,且具有较高数值精度. 相似文献
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By means of the theory of spline interpolation in Hilbert spaces, the bivariate polynomial natural spline interpolation to scattered data is constructed. The method can easily be carried out on a computer, and parallelly generalized to high dimensional cases as well. The results can be used for numerical integration in higher dimensions and numerical solution of partial differential equations, and so on. 相似文献
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In this paper, a numerical method for singular initial value problems of the Lane–Emden type in the second-order ordinary differential equations is proposed. The method changes solving the equation to solving a Volterra integral equation. We have applied the improved Legendre-spectral method to solve Lane–Emden type equations. The Legendre Gauss points are used as collocation nodes and Lagrange interpolation is employed in Volterra term. The results reveal that the method is effective, simple and accurate. 相似文献
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On choosing “optimal” shape parameters for RBF approximation 总被引:1,自引:0,他引:1
Many radial basis function (RBF) methods contain a free shape parameter that plays an important role for the accuracy of the method. In most papers the authors end up choosing this shape parameter by trial and error or some other ad hoc means. The method of cross validation has long been used in the statistics literature, and the special case of leave-one-out cross validation forms the basis of the algorithm for choosing an optimal value of the shape parameter proposed by Rippa in the setting of scattered data interpolation with RBFs. We discuss extensions of this approach that can be applied in the setting of iterated approximate moving least squares approximation of function value data and for RBF pseudo-spectral methods for the solution of partial differential equations. The former method can be viewed as an efficient alternative to ridge regression or smoothing spline approximation, while the latter forms an extension of the classical polynomial pseudo-spectral approach. Numerical experiments illustrating the use of our algorithms are included. 相似文献
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K. J. In 't Hout 《BIT Numerical Mathematics》2001,41(2):322-344
This paper deals with the adaptation of Runge—Kutta methods to the numerical solution of nonstiff initial value problems for delay differential equations. We consider the interpolation procedure that was proposed in In 't Hout [8], and prove the new and positive result that for any given Runge—Kutta method its adaptation to delay differential equations by means of this interpolation procedure has an order of convergence equal to min {p,q}, where p denotes the order of consistency of the Runge—Kutta method and q is the number of support points of the interpolation procedure.This revised version was published online in October 2005 with corrections to the Cover Date. 相似文献
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In general, we will use the numerical differentiation when dealing with the differential equations. Thus the differential equations can be transformed into algebraic equations and then we can get the numerical solutions. But as we all have known, the numerical differentiation process is very sensitive to even a small level of errors. In contrast it is expected that on average the numerical integration process is much less sensitive to errors. In this paper, based on the Sinc method we provide a new method using Sinc method incorporated with the double exponential transformation based on the interpolation of the highest derivatives (SIHD) for the differential equations. The error in the approximation of the solution is shown to converge at an exponential rate. The numerical results show that compared with the exiting results, our method is of high accuracy, of good convergence with little computational efforts. It is easy to treat nonhomogeneous mixed boundary condition for our method, which is unlike the traditional Sinc method. 相似文献