MAT 514 Project 3

Project Description

The goal of this project is to compute the inverse Laplace transform of the function
F(s) = 1/sqrt(s^3+m)
where m is the first digit of your SUID. The methods of MAT 514 do not work here, and WolframAlpha cannot help either. Instead, we can use Post's inversion formula which gives f(t) as a limit of a certain expression with k^th derivative of F, as k tends to infinity. In theory, choosing a larger value of k will produce a better approximation to the inverse transform. In practice, increasing k may lead to loss of precision. You should find the right balance (a suitable value of k) by trial and error.

For this project I recommend using Sage, powerful open-source mathematics software. To run it in a browser, follow this link: Sage Notebook and create an account. Sage can be also downloaded from sagemath.org and installed locally, but this is not as easy.

Technical Details

Here is a possible Sage input, with comments in italic:

s,t = var('s t')
declares variables s and t

F(s) = 1/sqrt(s^3+1)
use your value of m instead of 1

k = 1
you will need a larger value of k to get a reasonable approximation

dF = F.diff(k)
computes the k^th derivative of F

f(t) = (-1)^k/factorial(k)*(k/t)^(k+1)*dF(k/t)
Post's inversion formula

plot(f,0,10)
plots the approximate inverse on [0,10]

The result of your investigation should be submitted as a screenshot via Blackboard. You should also comment on your choice of k in the text area Comments. The project is due at 10PM on Friday, November 18.

Notes:
(1) Press Shift-Enter to evaluate a Sage command or group of commands.
(2) You can Interrupt and/or Restart a notebook in case of technical difficulties.
(3) Sage is written in Python, so you can use Python syntax such as "for k in range(a,b):" to jazz things up.

Optional reading: Life and work of Emil Post Wikipedia article on Sage