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基于复化Gauss-Legendre积分的Laplace变换数值反演及其应用

李小龙   

  1. 桂林航天工业学院机械工程学院, 桂林 541004
  • 收稿日期:2020-03-06 出版日期:2021-09-15 发布日期:2021-09-17
  • 通讯作者: 李小龙,lixiaolong20@gscaep.ac.cn

李小龙. 基于复化Gauss-Legendre积分的Laplace变换数值反演及其应用[J]. 数值计算与计算机应用, 2021, 42(3): 263-275.

Li Xiaolong. THE NUMERICAL INVERSION OF LAPLACE TRANSFORM BASED ON COMPLEX GAUSS-LEGENDRE QUADRATURE AND ITS APPLICATION[J]. Journal on Numerica Methods and Computer Applications, 2021, 42(3): 263-275.

THE NUMERICAL INVERSION OF LAPLACE TRANSFORM BASED ON COMPLEX GAUSS-LEGENDRE QUADRATURE AND ITS APPLICATION

Li Xiaolong   

  1. College of Mechanical Engineering, Guilin University of Aerospace Technology, Guilin 541004, China
  • Received:2020-03-06 Online:2021-09-15 Published:2021-09-17
Laplace变换的数值反演是一个病态问题.采用代数精度较高的数值积分近似Laplace变换截断积分,合理选取复平面上样本点以形成离散线性代数方程组是解决这个问题的途径之一.本文采用代数精度较高的复化Gauss-Legendre数值积分近似Laplace变换截断积分,推导了一种Laplace变换数值反演算法.其间,对于所形成的条件数很大的线性方程组采用基于约化奇异值分解的最小二乘法进行求解,以尽可能降低数值解的误差.使用该算法对简单测试算例进行数值反演,并将其结果与精确解进行对比,结果表明,相比经典的Gaver-Stehfest方法和基于Gauss-Legendre积分的方法,本文推导的反演算法可以达到满意的数值精度.同时,结合该算法采用半解析半数值方法对一个较为复杂的冲击凿岩问题的数值反演结果也表明该数值反演算法具有一定的实用性.
Numerical inversion of the Laplace transform is an ill-conditioned problem. Applying high order numerical quadratures to truncated integral in Laplace transform, then selecting positive real points on the complex plane suitably to form discretized linear algebraic equations is one kind of methods to solve it. The complex Gauss-Legendre integral with higher algebraic accuracy is adopted to approximate the truncated integral in this transform, then a new numerical inverse method is developed, in which the linear algebraic equations with large condition number is solved by the least squares method based on reduced singular value decomposition to eliminate the numerical error as many as possible. The comparisons between numerical inversion results and analytical solution for simple test example show that the proposed inversion method has the highest accuracy among the proposed, the Gaver-Stehfest method and the method based on Gauss-Legendre quadrature. In addition, semi-analytical and semi-numerical method is used to solve a little complex problem in the field of percussive drilling combining with the proposed numerical method above, which also shows the practicability for the method to some extent.

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