特解边界元法数值解三维Pennes方程及其应用

应用特解边界元法对非稳态的三维Pennes方程求解,将解分为一满足泊松方程的通解与一特解之和,通解按照边界元法求解,特解用分离变量法给出,与位置有关的部分采用截断多项式.逐时间段计算边界元的温度及热流值,然后计算域内点的值.并将该方法用于热疗温度场的数值实例计算.     

用松弛法解薛定谔方程

求解定态薛定谔方程常常会涉及到常微分方程的本征值问题.目前解常微分方程本征值用的比较多的方法是以龙格-库塔方法为基础的打靶方法.打靶方法常用,但是计算时间长.当边界条件比较复杂或比较敏感的时候,用松弛法会有更好的效果.本文用松弛法解薛定谔方程,并和理论解进行比较.发现松弛法得到的数值解和理论解符合度很高,而且使用松弛法...     

Application of He's parameter-expansion method for oscillators with smooth odd nonlinearities

This Letter applies He's parameter-expansion method to oscillators with smooth nonlinearities. The method does not depend upon small parameter assumption, hence it is very better than the perturbation method. In parameter-expansion method the solution and unknown frequency of oscillation are expanded in a series by a bookkeeping parameter. By imposing the non-secularity condition at each order in the expansion the method provides different approximations to both the solution and the frequency of oscillation. One iteration step provides an approximate solution which is valid for the whole solution domain. The method can be easily extended to other nonlinear oscillations.     

一类非线性方程类孤波的近似解法

采用了一个简单而有效的技巧,研究了一类扰动发展方程.首先引入求解一个相应典型方程的类孤波近似解,然后利用泛函映射方法得到了原扰动发展方程的近似解,指出了近似解级数的收敛性,并用解析方法,讨论了近似解的精度.     

On the solution of a vectorial radiative transfer equation in an arbitrary three-dimensional turbid medium with anisotropic scattering

The authors developed a numerical method of the boundary-value problem solution in the vectorial radiative transfer theory applicable to the turbid media with an arbitrary three-dimensional geometry. The method is based on the solution representation as the sum of an anisotropic part that contains all the singularities of the exact solution and a smooth regular part. The regular part of the solution could be found numerically by the finite element method that enables to extend the approach to the arbitrary medium geometry. The anisotropic part of the solution is determined analytically by the special form of the small-angle approximation. The method development is performed by the examples of the boundary-value problems for the plane unidirectional and point isotropic sources in a turbid medium slab.     

Closure approximation error in the mean solution of stochastic differential equations by the hierarchy method

The iterative or stochastic Green's function method of solution of stochastic differential equations is used to find the error terms in the solution and mean solution due to truncation in the hierarchy method. A comparison is made of solutions by the iterative and the hierarchy method.Supported by a grant from the Sloan Foundation.     

Propagation of finite amplitude sound waves radiated from a pulsating sphere

The propagation of weakly non-linear acoustic waves is studied, which are radiated from a harmonically pulsating sphere in an inviscid perfect gas. A representation of the solution is presented for a far field equation of the first order, which is closely related to the solution obtained by the method of renormalization. The applicability of the method to the present problem is proved within the first order approximation. The asymptotic solution in the far field is obtained up to the second order by using the method of renormalization. A uniformly valid solution is, to the second approximation, constructed by matching the near field solution with the far field solution. Also investigated is the effect of dimensionless parameters and the second order correction on the acoustic shock formation distances and the non-linear distortion of waveforms.     

A new similarity solution of the Burgers equation with linear damping

The direct method of Clarkson and Kruskal is used to study the symmetry reductions of the Burgers equation with linear damping. The classical similarity reductions reported previously is recovered. The new, nonclassical similarity reduction is obtained. The new similarity solution is given, and it is also obtained by means of the singular manifold method. The nonclassical method is used to demonstrate that the new exact solution is indeed a nonclassical similarity solution. This work has been supported by the Postdoctoral Science Foundation of China.     

Numerical solution of Helmholtz equation of barotropic atmosphere using wavelets

The numerical solution of the Helmholtz equation for barotropic atmosphere is estimated by use of the wavelet-Galerkin method. The solution involves the decomposition of a circulant matrix consisting up of 2-term connection coefficients of wavelet scaling functions. Three matrix decompositions, i.e. fast Fourier transformation (FFT), Jacobian and QR decomposition methods, are tested numerically. The Jacobian method has the smallest matrix-reconstruction error with the best orthogonality while the FFT method causes the biggest errors. Numerical result reveals that the numerical solution of the equation is very sensitive to the decomposition methods, and the QR and Jacobian decomposition methods, whose errors are of the order of 10^{-3}, much smaller than that with the FFT method, are more suitable to the numerical solution of the equation. With the two methods the solutions are also proved to have much higher accuracy than the iteration solution with the finite difference approximation. In addition, the wavelet numerical method is very useful for the solution of a climate model in low resolution. The solution accuracy of the equation may significantly increase with the order of Daubechies wavelet.     

一个全球气候非线性振荡模型的近似解

研究了一个全球气候的非线性模型. 利用摄动理论和方法, 构造了相关问题解的渐近展开式, 该问题解的渐近展开式具有较高的近似度. 摄动渐近方法是一个解析方法, 得到的解还能够继续进行解析运算. 关键词： 非线性 摄动 全球气候     

A hybrid discontinuous Galerkin method for computing the ground state solution of Bose–Einstein condensates

A numerical method for computing the ground state solution of Bose–Einstein condensates modeled by the Gross–Pitaevskii equation is presented. In this method, the three-dimensional computational domain is divided into hexahedral elements in which the solution is approximated by a sum of basis functions. Both polynomial and plane wave bases are considered for this purpose, and Lagrange multipliers are introduced to weakly enforce the interelement continuity of the solution. The ground state is computed by an iterative procedure for minimizing the energy. The performance results obtained for several numerical experiments demonstrate that the proposed method is more computationally efficient than similar solution approaches based on the standard higher-order finite element method.     

Homotopic mapping solving method of the reduces equation for Kelvin waves

A reduces equation of the Kelvin wave is considered. By using the homotopic mapping solving method, the approximate solution is obtained. The homptopic mapping method is an analytic method, the obtained solution can analyse operations sequentially.     

Convergence of fourier methods for Navier-Stokes equations

The Fourier-Galerkin method is used to simulate fluid flows in two and three dimensions, on domains with periodic boundary conditions. It is proved that the numerical solution converges towards the solution of Navier-Stokes equations. The rate of convergence depends on the smoothness of the mathematical solution. Finally, it is shown that the Fourier-Galerkin method can be interpreted as a projection method. This observation may lead to more sophisticated convergence proofs.     

三维赤道海气振子模型的近似解

研究了一个赤道Kelvin波的方程. 利用变分迭代解法, 得到了其近似解. 变分迭代方法是一个解析方法, 得到的解还能够继续进行解析运算. 关键词： 变分迭代 近似解 Kelvin     

Numerical treatment of acoustic problems with boundary singularities by the singular boundary method

The singular boundary method (SBM) is a novel boundary-type meshless method based on the fundamental solution of the given governing equation. The SBM employs the origin intensity factors to circumvent the singularities resulting from the fundamental solutions. In this paper, we investigate the acoustic problems with boundary singularities using the SBM. This is achieved by combining the SBM with the singularity subtraction techniques where the solution is decomposed into the singular solution and the regular solution. The singular solution is derived analytically which satisfies the governing equation and the corresponding boundary conditions containing the singularities. Then the regular solution is obtained by the SBM. Numerical examples show the excellent performance of the proposed technique.     

Various Kinds Waves and Solitons Interaction Solutions of Boussinesq Equation Describing Ultrashort Pulse in Quadratic Nonlinear Medium

The consistent tanh expansion (CTE) method is applied to the (2+1)-dimensional Boussinesq equation which describes the propagation of ultrashort pulse in quadratic nonlinear medium. The interaction solutions are explicitly given, such as the bright soliton-periodic wave interaction solution, variational amplitude periodic wave solution, and kink-periodic wave interaction solution. We also obtain the bright soliton solution, kind bright soliton solution, double well dark soliton solution and kink-bright soliton interaction solution by using Painlevé truncated expansion method. And we investigate interactive properties of solitons and periodic waves.     

Calculation of the speed of sound in reciprocal systems from the known ion concentrations

A method of calculating the deviations of the speed of sound in a mixed solution containing two types of cations and two types of anions from the speed of sound in pure water by using the known ion concentrations in the solution and the concentration dependences of the speed of sound in binary solutions of the salts entering the mixed solution is proposed. The method is based on the ionic strength rule and the equiprobable ion distribution hypothesis. The calculated values of the deviations of the speed of sound in a mixed solution from the speed of sound in pure water (for solution temperatures of 15 and 25°C) are in fairly good agreement with experimental data. The proposed method is applied for calculating the speed of sound in seawater.     

(2+1)维扰动时滞破裂孤波方程行波解的摄动方法

研究了一类(2+1)维扰动时滞破裂孤波方程. 首先讨论了对应的无时滞情形下的破裂方程,利用待定系数投射方法得到了孤波精确解. 再利用同伦、摄动近似方法得到了扰动破裂孤波方程的行波渐近解. 关键词： 孤波 行波解 近似解     

基于小波变换的最小二乘相位解缠算法

最小二乘法是求解二维相位解缠问题最稳健的方法之一,其本质是在最小二乘意义下使缠绕相位的离散偏导数与解缠相位的偏导数整体偏差最小,并等效为可求解一大型的稀疏线性方程系统。由于系统矩阵结构的稀疏性,在采用迭代法求解时收敛速度非常慢。为了改善收敛特性,提出一种基于多分辨率表示的离散小波变换相位解缠算法。利用小波变换将原线性系统转化成具有较好收敛条件的等价新系统。仿真实验表明,该方法能够很好的恢复真实相位,其解缠效果优于Gauss-Seidel松弛迭代和多重网格法。     

微重力下圆管毛细流动解析近似解研究

应用同伦分析法研究微重力环境下圆管毛细流动解析近似解问题, 给出了级数解的表达公式. 不同于其他解析近似方法, 该方法从根本上克服了摄动理论对小参数的过分依赖, 其有效性与所研究的非线性问题是否含有小参数无关, 适用范围广. 同伦分析法提供了选取基函数的自由, 可以选取较好的基函数, 更有效地逼近问题的解, 通过引入辅助参数和辅助函数来调节和控制级数解的收敛区域和收敛速度, 同伦分析法为圆管毛细流动问题的解析近似求解开辟了一个全新的途径. 通过具体算例, 将同伦分析法与四阶龙格库塔方法数值解做了比较, 结果表明, 该方法具有很高的计算精度. 关键词： 圆管 微重力 毛细流动 同伦分析法