Approximate Inversion of the Laplace Transform
Approximate Inversion of the Laplace Transform
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发表时间：
1998
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通讯作者：
A. Cheng;P. Sidauruk
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作者：
A. Cheng;P. Sidauruk
常用于求解线性偏微分方程以消除时间维度。通过解析技术[Detournay and Cheng 19881]或数值方法(如有限元或边界元方法]Cheng and Ou 1989；Cheng and Detournay 19881。这样得到的解需要倒转到时域。精确的反演通常很难进行，因此采用近似反演技术。有许多近似的拉普拉斯反演算法。（有关全面的参考书目，请参见[Piessens 1977.5; Piessens and Dang 19761]。）考虑到计算效率和准确性的问题，已经测试和比较了一些技术[成本1964；Davies和Martin 1979；Narayanan and Beskos 19821。根据文献中的建议和我们自己在工程应用中的经验，我们在这里介绍数学包NLaplaceInversion。m，实现了五种反演算法，分别对应Davies和Martin[19791]分类的五种类别。重要的是要强调“拉普拉斯变换的逆在合理的扰动下是不稳定的”[Bellman et al. 19661]。这可以通过考虑函数f(t) = sinat, a > 0，它的拉普拉斯变换f(s) = a/(a2 + s2)来看出。F(s)可以通过取足够大的值而任意减小，而F(t)的逆变换在-1和1之间振荡。因此，不可能设计出一种对所有类型的函数都能很好地执行的通用算法
is often used in the solution of linear partial differential equations to eliminate the time dimension. The resulting systems become easier to solve by analytical techniques [Detournay and Cheng 19881, or numerical methods such as finite element or boundary element methods [Cheng and Ou 1989; Cheng and Detournay 19881. The solutions thus obtained need to be inverted to the time domain. The exact inversion is normally difficult to carry out, so approximate inversion techniques are used. There are many approximate Laplace inversion algorithms. (For comprehensive bibliographies, see [Piessens 197.5; Piessens and Dang 19761.) With issues of computational efficiency and accuracy in mind, a handful of techniques have been tested and compared [Cost 1964; Davies and Martin 1979; Narayanan and Beskos 19821. Based on recommendations in the literature and our own experience in engineering applications, we present here the Mathematics package NLaplaceInversion .m, which implements five inversion algorithms, one in each of the five categories classified by Davies and Martin [ 19791. It is important to emphasize that “the inverse of the Laplace transform is not stable under reasonable perturbations” [Bellman et al. 19661. This can be seen by considering the function f(t) = sin at, a > 0, and its Laplace transform F(s) = a/( a2 + s2). While F(s) can be made arbitrarily small by taking a sufficiently large, the inverse transform f(t) oscillates between -1 and 1. It is therefore impossible to devise a universal algorithm that performs well for all types of func-