MA1035: Engineering Modeling using Dynamic Systems

Session 09: The Laplace Transform

Prof. Alejandro Ucan-Puc

Tecnológico de Monterrey • Departamento de Ciencias

Goals:

  • Motivate the Laplace transform.

  • Define the Laplace transform.

  • Compute the Laplace transform of some functions.

Integral transforms

The derivative and anti-derivative are transformations of functions, and can be thought of as linear transformations.





Remember that through anti-derivatives we can define functions.

Motivation:

Some problems (stated in some domain) are difficult to solve in their original presentation, but transformed can be simpler. The integral transform maps an equation in its original "domain" into another "domain" in such a way that the manipulation of the equation is simpler than originally and the solution can be reconverted to the original domain.

Laplace Transform

Definition: the Laplace transform of a function $f(t)$ with $t\geq 0$ is the function $F(s)$ defined by $$F(s)=\int_0^\infty f(t)e^{-st}dt=\lim_{b\to\infty}\int_0^b f(t)e^{-st}$$ where $s$ is a parameter.

The usual notation for the Laplace transform is $\mathcal{L}\{f(t)\}=F(s).$

Example:

Compute the Laplace transforms of $f(t)=1$ and $f(t)=t.$

Example:

Compute the Laplace transform of: $f(t)=-3t^2+2t+1,$ $f(t)=\cos(3t)$ and $f(t)=e^{2t}.$

The advantages

The Laplace transform calculation is something that you do only once, because there are some formulaes that you can use directly.



Theorem: It follows that:

  • $\mathcal{L}\{1\}=\frac{1}{s}.$
  • $\mathcal{L}\{t^n\}=\frac{n!}{s^{n+1}}.$ for an integer $n>0$
  • $\mathcal{L}\{e^{at}\}=\frac{1}{s-a}.$
  • $\mathcal{L}\{\sin(kt)\}=\frac{k}{s^2+k^2}.$
  • $\mathcal{L}\{\cos(kt)\}=\frac{s}{s^2+k^2}.$

Laplace transform Linearity

Linearity: Let $f$ and $g$ be two functions defined on $[0,\infty)$ such that their Laplace transforms exist, then $$\mathcal{L}\{c_1 f+c_2 g\}=c_1\mathcal{L}\{f\}+c_2 \mathcal{L}\{g\}.$$

Example

Using Linearity in the Laplace transform, transform the following functions:

  • $f(t)=-5+3t+4t^2$
  • $f(t)=5e^{t}-2e^{-3t}$
  • $f(t)=-3t^3+\cos(4t)+ 2\sin(4t).$
  • $f(t)=\frac{\sin(3t)-\cos(4t)}{5}.$

Laplace Transforms for Piecewise Functions:

Using the integral definition, compute the Laplace transform of the following functions:

  • $f(t)=-1$ if $0\leq t \leq 1$ and $f(t)=1$ if $t>1.$
  • $f(t)=0$ if $0\leq t\leq a$ and $f(t)=t$ if $t>a.$
  • $f(t)=1$ if $a\leq t\leq b$ and $f(t)=0$ otherwise.

Summary & Next Step

Integral transform concept, formal definition, Laplace transforms of elementary functions, algebraic linearity, and piecewise forcing functions.

Next Session: Applied Laplace Transform & IVPs