InverseLaplaceTransform problems

*To*: mathgroup at smc.vnet.net*Subject*: [mg19325] InverseLaplaceTransform problems*From*: techie at mit.edu*Date*: Mon, 16 Aug 1999 02:14:56 -0400*Organization*: Massachvsetts Institvte of Technology*Sender*: owner-wri-mathgroup at wolfram.com

Hi, I am having several difficulties with (Inverse)LaplaceTransform under Mathematica 3.0.1 for Linux, with the SignalProcessing 3.0 package. I have tried both sets of Laplace transforms located in the calculus package (part of Mathematica) and the SignalProcessing 3.0 package; neither one is sufficient for my needs, and I need help getting either (or both) to work properly. Here is the description of the problems: 1. I have found that the (Inverse)LaplaceTransform functions in the Signals package are less useful than the ones found in Calculus`LaplaceTransform. A simple example would be (for the Signals package): InverseLaplaceTransform[LaplaceTransform[v[t], t, s], s, t] yields: (1/362880)(362880DiracDelta[0]DiracDelta[t]v[0] + 362880 v[0] DiracDelta'[0]DiracDelta'[t] - ..... (about 22 more lines!) instead of the desired result: v[t]! This is such a simple rule, you'd think that it would be standard. For the standard LaplaceTransform: <<Calculus`LaplaceTransform` InverseLaplaceTransform[LaplaceTransform[v[t], t, s], s, t] yields -> v[t] as expected. Also, InverseLaplaceTransform[s LaplaceTransform[v[t], t, s], s, t] yields -> DiracDelta[t] v[0] + v'[t] as expected. 2. However, if the expression to be InverseLaplaceTransform'ed instead contains a polynomial fraction function of s (for example: s / (s + 1) ) times the LaplaceTransform of some variable v[t], for example: InverseLaplaceTransform[(s / (s+1)) LaplaceTransform[v[t], t, s],s, t] yields -> \!\(InverseLaplaceTransform[\(s\ LaplaceTransform[v[t], t, s]\)\/\(1 + s\), s, t]\) which is the same! It instead should be: v[t] * (-E^(-t) + DiracDelta[t]), where "*" denotes the convolution integral from 0 to t. This can be simplified further to: v[t] \!\(\(-\(\[Integral]\_0\%t\( E\^\(\(-t\) + \[Tau]\)\ v[\[Tau]]\) \[DifferentialD]\[Tau]\)\)\) I would really like to find a way to get this to work since I am trying to solve systems of circuit equations in the Laplace domain, then inverse transform to get the time domain answer: e.g. Y[s] = H[s] X[s] Or, equivalently in the time domain: y[t] = h[t] * x[t] h[t] is the InverseLaplaceTransform of H[s]. (where H[s] is the system function having a polynomial numerator and denominator in s; h[t] is the impulse response of the system; x[t] is the input to the system; y[t] is the output). I just wanted to check to see if anyone out there has already solved this problem / deficiency in the LaplaceTransform functions. thanks a lot for any suggestions / helpful hints, etc. Ed Ouellette techie at mit.edu