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WindowTitle->17. Numerical Inverse Laplace Transform | Laplace Transforms in \
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In this chapter, numerical methods for the inverse Laplace transform will be \
discussed. Numerical methods can prove useful for functions for which \
obtaining an analytic inversion is either impossible or too challenging.\
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problem. If two or more methods yield close results, one can have more \
confidence in the numerical answer. Wolfram Language contains the following \
methods for numerical inversion: Stehfest, Weeks, Papoulis, Durbin, \
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with numerous examples illustrating their applications. For more details on \
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The following indicates a small error between the numerical and analytical \
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Consider the Weeks (1966) method, which is based on the expansion of the \
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The small error between the exact (obtained using the previous analytic \
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Plot the numeric (via the Weeks method) inverse and the analytic inverse \
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As you can see, the numerical answer is inaccurate. To enhance accuracy, \
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Consider the Papoulis (1957) method, which is based on the expansion of the \
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The small error between the exact (obtained using the previous analytic \
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Notice that the Laplace inversion with all three methods yields numerical \
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The error between the exact (obtained using the previous analytic solution) \
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Plot the numeric (via the Durbin method) inverse and the analytic inverse \
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The error between the exact (obtained using the previous analytic solution) \
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 "The numerical methods discussed so far are based on the direct numerical \
evaluation of the inversion integral along the vertical (Bromwich) line: ",
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 ", it has proven advantageous to perform analytic inversion by deforming the \
vertical line into the left half-plane. This suggests the possibility of \
attempting numerical inversion using a similar approach. In the following \
discussion, two methods\[LongDash]the Talbot method and the Piessens method\
\[LongDash]will be introduced. It is important to note that when deforming \
the contour to the left, care must be taken to avoid crossing any \
singularities. Otherwise, the integral along such a contour will no longer \
accurately represent the original integral along the vertical line."
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The two methods effectively differ in their choices of contours. The Talbot \
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The error between the exact (obtained using the previous analytic solution) \
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Plot the numeric (via the Talbot method) inverse and the analytic inverse \
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Cell["\<\
For all the examples discussed thus far, the analytic inverse has been known. \
This was convenient for crosschecking the answer obtained using various \
numerical methods. However, the primary motivation for numerical inversion \
lies in the reality that analytic inversion is not always possible. The only \
feasible means to acquire insights into the time-domain function, in such \
cases, is through the utilization of numerical techniques. \
\>", "Text",
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The error between the exact (obtained using the previous analytic solution) \
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Plot the numeric (via the Weeks method) inverse and the analytic inverse \
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The small errors between the exact (obtained using the previous analytic \
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The small errors between the exact (obtained using the previous analytic \
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Plot the numeric (via the Piessens method) inverse and the analytic \
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