The Laplace Transform

Suggested Viewing

Khan Academy has a great video tutorial series on the Laplace Transform. It is suggested that you watch this series of 19 videos before attempting this module.

Laplace Transform 1: Introduction to the Laplace Transform

Brief History

The Laplace Transform is credited to French mathematician and astronomer Pierre-Simon Laplace. Well before the work of Laplace, however, mathematical genius Leonhard Euler had studied integrals in the form \( \int_0^{\infty} f(t)\,e^{-at}\mathrm{d}t \) and \( \int f(x)\,x^{A}\,\mathrm{d}x \) as solutions of differential equations, but did not pursue this topic very far. Joseph Louis Lagrange, Italian mathematician and astronomer who succeded Euler as the director of mathematics at the Prussian Academy of Sciences in Berlin, began to study integrals in the form \( \int_0^{\infty} f(t)\,e^{-at}\,a^{t}\,\mathrm{d}x \) in connection with his work on integrating probability density functions. Laplace was the next person to seriously work on this topic, and took a critical step forward by applying the idea of a "transformation" rather than just looking for a solution in the form of an integral. He looked for soltions with the following equation: \( \int x^{s}\,\phi (x)\,\mathrm{d}x = F(s) .\) In 1809, Laplace extended his transform to find solutions that diffused indefinitely - giving us our popular Laplace transformation. Although the results had been published for at least 70 years, the transformation was not given a true physical meaning until Oliver Heaviside came up with the same equation on his own in the 1880s. Oliver Heaviside was an English mathematician, electrical engineer, and physisict (completely self-taught) who was working on the solution to multi-degree differential equations for electrical systems in 1880s. These kinds of equations (usually with 10 or more derivatives of y) would usually take days or week for most people to solve. Heaviside was able to solve these kinds of equations within hours. Single-handedly, between 1880--1887, he invented operational calculus -- a new method using operator \( \texttt{D} \) notation that allowed him to transfer difficult differential equations into simple algebraic equations. By using his own version of the Laplace Transform, he would apply his operation to the differential equation and then apply his own Inverse Laplace Equation to obtain the answer. It took many years for academia to accept his results because a lack of a rigorous proof for his methods. He replied to this criticism with the famous statement "Mathematics is an experimental science, and definitions do not come first, but later on," claiming that "I do not refuse my dinner simply because I do not understand the process of digestion." To this day, the Inverse Laplace Transform is the most difficult process to understand when solving differential equations in this method.

Concept Overview

Essentially, we will be doing what Heaviside did - we will take a differential equation, use differential operator notation to create an algebraic equation that can be solved for the Laplace Transform of "y" (our dependent variable), apply the Laplace Transform, then apply the Inverse Laplace Transform to obtain the solution. We again will be using Initial Value Problems (IVP)/Cauchy Problems with specified initial conditions to give us a unique solution. Thus, our answer will also be unique when using the Laplace Transform method - we will have no arbitrary constants. Something to note, however, is that this method is ONLY valid for "t" (our independent variable) when "t" is > 0. We will be using a Heaviside function that will allow our answer to maintain these conditions - (the Heaviside function will be more well defined in the section labled "Heaviside Functions")

1. Differential Equation $y'' + y' + ...\Rightarrow$ Apply Laplace Transform
2. Algebraic Equation $\mathcal{L}[y] = \lambda^2 + 3 \lambda... \Rightarrow$ Apply Inverse Laplace Transform
3. Answer, defined for t > 0 $te^{2t}+4\sin(3t)+... \Rightarrow$ Apply Heaviside
4. Final Answer $H(t)*(te^{2t}+4\sin(3t)+...)$

Formal Definition

Suppose $f(t)$ is either a continuous function (or more realisitically) a piecewise continuous function.

Notice that this integral, defined for t from 0 to infinity becomes an a new equation of \( F(\lambda ) \) -- this is our algebraic equation.
The next few sections will lead you through a more rigorous introduction to the Laplace transform, and it's applications in MuPAD.

Other Great Resources for the Laplace Transform

Check out Paul's Online Math Notes from Lamar University:
Laplace Transform