# Symbolic Inverse Laplace Transform Applet

## Features

• User defined arbitrary rational polynomial using string expression
• Automatic calculation of symbolic form of the partial fraction expansion
• Plot of the inverse Laplace transform for the domain of t>0

## Rules and Theories

A rational polynomial is defined by where the numerator N(s) and the denominator M(s) are polynomials with real-valued coefficients. For practical purposes, the degree of N(s), n, is assumed to be less than that of M(s), m. The key to perform the partial fraction expansion is the find the m poles. Denote to be the j-th distinct complex root of M(s) with multiplicity . Then

According to Heaviside's expansion theorem,

The challenges are:

• Poles are in general complex valued
• The evaluation of the expansion coefficients requires the ability to take derivatives of a polynomial with complex coefficients

The above issues are handled with Java's OOP approach.

## In the Laplace transform domain, suppose X(s), Q(s), and Y(s) represent the input signal, the system transfer function and the output signal, respectively. In general, the input/output relation is H(s) = Y(s) = X(s) Q(s). This applet can be used to find the time-domain response y(t) for any known input signal and transfer functions, provided Y(s) can be expressed in form of a rational polynomial. There are two notable special cases: (i) For an impulse excitation, X(s) = 1, then letting H(s) = Y(s) = Q(s) will lead to the result of the impulse response. (ii) For a step function input, X(s) = 1/s, then letting H(s) = Q(s)/s will lead to the step-function response.

• The user need to enter strings for both the numerator and the denominator polynomials
• Simple polynomial term can be entered with free-hand written forms, e.g., s^6-7s^2+2
• Factored terms can be entered using parentheses, e.g., (s)(s+1)(s^2+3s-6)
• A click on "Inverse Laplace Transform" button triggers the calculation. The resultant symbolic expression for the inverse Laplace transform is displayed in the text area below the "h(t)=" label
• The user can assign the upper bound in the time-domain, tmax, and the number of sampling points, N
• A click on the "Plot h(t)" button will cause a plot of the time-domain response function. NOTE: If the option "parse h(t)" is checked. The function h(t) as defined in the text-box will be parsed. User can use this feature to plot any function h(t), which can be unrelated to the inverse Laplace transform at all. Otherwise, no matter what content appears in the text-box, the numerical value of the inverse Laplace transform of the original rational polynomial will be plotted.

## Validation with Known Transform Pairs

This applet has been tested with selected examples for perfect agreement..

## References

M. E. Van Valkenburg, "Network Analysis", 2nd Edition, Prentice Hall, Englewood Clifs, 1964.