Laplace变换是求解偏微分方程的有效手段, 但Laplace逆变换的数值求解一直是一项难题. 本文引入梯形公式的指数收敛性质、介绍了该性质结合Hankel围道高精度数值求解Laplace逆变换的方法. 随后以抛物型围道为例, 深入分析了该方法在多种情况下的各类误差、提出了系统性的最优参数选取方法, 并将其应用于热方程的求解中. 经过数值检验, 无论是直接求解Laplace逆变换, 还是结合谱方法求解热问题, 本文提出的参数选取方法都能达到机器精度.

Laplace transform is an effective method for solving partial differential equations, but the numerical solution of Laplace inverse transform has always been a challenge. This article introduces the exponentially convergent trapezoidal rule and introduces the method of combining this with Hankel contour to solve the Laplace inverse transformation with high accuracy. Subsequently, taking the parabolic enclosure as an example, the various errors of this method in various situations were analyzed in depth, and a systematic optimal parameter selection method was proposed, which was applied to solve the heat equation. After numerical testing, whether directly solving the Laplace inverse transform or combining spectral methods to solve heat problems, the parameter selection method proposed in this article can achieve machine accuracy.