一维波动方程第二类边界条件混合问题的初等函数近似解

P. G. Lasy
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引用次数: 0

摘要

研究一类一维波动方程具有第二类边界条件的混合问题。这个问题的解是用格林函数写成积分形式的。在实际应用中,这种解的用处不大,因为,首先,格林函数是一个三角级数,因此,它的计算有一定的困难;其次,问题的解中必须近似地计算包含格林函数的五个积分;第三,解的近似计算的误差是极难估计的。本文克服了这些困难,即用周期分段线性函数找到了格林函数的简单表达式,用周期分段线性函数、分段二次函数和分段三次函数计算了近似解中包含的积分,最后得到了一个简单有效的近似误差估计。误差估计在问题的网格步长上是线性的,在任意固定的时间点上在空间变量上是均匀的。这样,具有任意小误差的问题的近似解可以有效地用初等函数表示。给出了用该方法求解该问题的一个实例,并绘制了精确解和近似解的图形。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Approximate Solution Using Elementary Functions of Mixed Problem with Boundary Conditions of the Second Kind for One-Dimensional Wave Equation
The paper considers a mixed problem with boundary conditions of the second kind for a one-dimensional wave equation. The solution to this problem is written in integral form using the Green’s function. For practical use, this solution is of little use, since, firstly, the Green’s function is a trigonometric series and, therefore, its calculation presents certain difficulties, secondly, it is necessary to calculate approximately the five integrals with the Green’s function included in the solution of the problem, and, thirdly, it is extremely difficult to estimate the error of the approximate calculation of the solution.  In this work, these difficulties are overcome, namely, simple expression for the Green’s function  is found in terms of a periodic piecewise linear function, the integrals included in the approximate solution are calculated using periodic piecewise linear, piecewise quadratic and piecewise cubic functions, and, finally,  a  simple and efficient estimate of the approximation error is obtained. The error estimate is linear in the grid steps of the problem and uniform in the spatial variable at any fixed point in time.  Thus, an approximate solution of the problem with an arbitrarily small error is effectively expressed in terms of elementary functions.   An example of solving the problem by the proposed method is given, and graphs of the exact and approximate solutions are plotted.
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