Differential equations may be expressed as an algebraic problem through the Laplace Transform. Moreover, a transfer function characterizes a system for analysis and is derived from the Laplace transform of the differential equation(s). However, in all cases the inverse Laplace transform is needed to find the time-domain solution. Analytical solutions exist for many problems, but a general numerical solver would be ideal. Then, specific electric or mechanical systems may be easily represented by transfer functions, and a numerical solver may produce the time-domain response of the system for various inputs.
Laplace Transforms
Before going into inverse Laplace transforms, fundamentally understanding Laplace Transforms may help. These videos conceptually and visually present the meaning of Laplace Transforms.
Inverse Laplace Transforms
These videos give a refresher on solving differential equations using Laplace Transforms. This involves finding the inverse Laplace Transform.
Transfer Functions
The following videos explain more about transfer functions specifically and how they are utilized. In either case, finding a solution in the time-domain necessitates calculating the inverse Laplace transform.
Numerical Methods of Inverse Laplace Transforms
Despite software like MATLAB having inverse Laplace transforms available, the topic is niche; YouTube does not have examples of people inversing Laplace transforms numerically. But, there are some papers.
Kathrin Spendier
In this informal paper by Kathrin Spendier, two methods are presented. One method is Fourier Series expansion, which uses a Bromwich contour inversion integral. Another method is Gaver-Stehfest, which only considers the real axis of the Laplace transformed function. The Gaver-Stehfest method works well for functions like Y(s)=1/(s+1), which has the Laplace inverse of y(t)=e^(-t).
This is to be expected since it is exclusively an exponential decay, which is what the x-axis in the Laplace transform space corresponds to. However, it does not work as well for functions that contain frequency components in the time-domain. Y(s)=1/(S^2+1), which has the Laplace inverse of y(t)=sin(t), progressively loses accuracy with increasing t after t=2.
Patrick O. Kano
In this presentation, Patrick O. Kano presents multiple insights and methods for Laplace inversion.
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