把常微分方程边值问题转化为积分方程,有个很重要的方法就是利用格林函数来求解.讨论了一类二阶线性常微分方程的边值问题,求出它在不同边值条件下的格林函数,从而给出这类方程格林函数的一般求解方法及其应用.

To transform the ordinary differential equation boundary value problem into integral equations , There is a very important method is solved by using the green''s function .The boundary value problems of a kind of second order linear ordinary differential equations are discussed in this paper .The paper considers the Green’ s functions for it with some boundary conditions .Then it refers to the general method of calculating and some applications of the Green ’ s function .

采用声模拟理论预测非紧致结构与非定常流动相互作用诱发的气动噪声时,为考虑边界对声场的散射影响,需要应用非紧致格林函数。为此,发展了一种基于边界元思想的非紧致格林函数数值计算方法,该方法适用于任意外形结构,能直接计算非紧致格林函数及其偏导数。以二维圆柱边界为算例,采用理论解析方法推导了非紧致格林函数及其偏导数的正确表达式,并将数值计算方法结果与理论解析方法结果相比较,验证了数值方法的正确性。

10.3969/j.issn.1006-1335.2013.02.002

通过寻找满足非紧致边界的精确格林函数，发展了一种基于声模拟理论的非紧致边界气动噪声数值预测方法。该方法首先采用边界积分方法计算满足相应边界条件的精确格林函数，然后采用精确格林函数求解FW-H方程，进行远场气动噪声预测。考虑了非紧致边界对声波传播的散射作用，适用于尺寸较大且几何外形复杂的边界与非定常流动相互作用诱发的气动噪声的数值预测。以雷诺数90000、马赫数0.2的圆柱绕流诱发的气动噪声为例，数值预测了非紧致边界条件下的气动噪声，并与采用自由空间格林函数求解FW-H的计算结果进行了对比，对该方法进行了验证。

A computational method was proposed to evaluate the aerodynamic noise generated by unsteady flow over non-compact boundary. Firstly, the boundary element method was applied to compute the exact Green’s Function that satis-fies the specific boundary conditions. Then, the numerical results of the exact Green’s Function were used to solve the FW-H equation to predict the far field noise. This numerical method can predict the radiation noise from the aerodynamic sources as well as the scattering noise from the non-compact surfaces. Especially, this approach is suitable for the boundaries with large sizes and complex geometric shapes. The present method was applied to predict aerodynamic noise generated by turbu-lent flows at Reynolds number of 90 000 and Mach’s number of 0.2 over a circular cylinder with non-compact boundary conditions. Results were compared with those of FW-H equations based on the free-space Green’s function to verify the ef-fectiveness of this method.

为了解决时域格林函数方法在计算非直壁浮体的数值发散问题，基于三维势流理论，对时域匹配直接边界元方法的数值特性进行了系统研究。该方法在流场中引入匹配控制面，将流场划分为内域和外域，内域中采用简单格林函数直接边界元方法，外域中采用满足线性自由面条件的时域格林函数方法，进行浮体时域水动力计算。研究结果表明：该方法外传波浪效率高，时域长期模拟结果稳定。基于同一网格模型，可以准确模拟不同波浪条件下浮体的时域水动力，无需重新划分网格，且网格数量大幅减少。

This paper presents an examination of the numerical characteristics of the time domain matching direct boundary element method, which is based on the three dimensional potential flow theories, to solve the numerical divergence problem of the time domain Green function method for calculating hydrodynamics of the non?sided?wall floating body. The control matching surface was introduced to flow field to decompose the flow field into the inner and outer domains. The simple Green function direct boundary element method was employed in the inner domain, and the time domain Green function method, which satisfies the linearized free surface condition, was applied in the outer domain. The time domain hydrodynamics of floating body was calculated. The results show the present method can transmit disturbing waves with high efficiency, and produce stable numerical results for longer simula?tions. Furthermore, based on the same grid model, the present approach can accurately evaluate the hydrodynam

研究一类分数阶微分方程的边值问题,通过常规方法将其转化为等价的积分方程,得到格林函数及其性质.

This paper studies boundary value problem of a class of fractional differential equations , which are transformed into an equivalent integral equation by use of the conventional method .Green function and its proper-ties are obtained .