Model Impulse Force with the Dirac Delta Function & Use the Laplace Transform, Generalized Function

After solving a discontinuous sinusoidally forced initial value problem using the unit step function and Mathematica, I then talk about how to mathematically model an impulse force using the Dirac Delta function, which is a generalized function, or distribution. The Laplace Transform of the Dirac Delta function is needed to analyze the model, though it must be compared with experimental results. As said before, the Dirac Delta function is an example of a generalized function, or distribution. Intuitively, its value is infinity at one point and zero elsewhere and we say that the integral of the Dirac Delta function equals one. Technically, it is defined by how it integrates against other functions, giving the value of that function at the point where the Dirac Delta function equals infinity. This defines a linear functional from a function space to the real numbers. The Laplace Transform of the Dirac Delta function that equals infinity at t = a is e^(-a*s). This can then be used to solve second order forced harmonic oscillator initial value problems where a constant multiple of the delta function is the forcing function.

The Theory of Distributions: A Nontechnical Introduction, by Ian Richards and Heekyung Youn: https://amzn.to/2M5Yjmk

Differential Equations, 4th Edition (by Blanchard, Devaney, and Hall): https://amzn.to/35Wxabr. Amazon Prime Student 6-Month Trial: https://amzn.to/3iUKwdP.

Bill Kinney's Differential Equations and Linear Algebra Course, Lecture 37B.

(a.k.a. Differential Equations with Linear Algebra, Lecture 37B, a.k.a. Continuous and Discrete Dynamical Systems, Lecture 37B).

#impulseforce #diracdelta #laplacetransform

Google drive link for Differential Equations and Linear Algebra course lecture documents: https://drive.google.com/drive/folders/1kM9ko27wA5NvmZ8CxHIKygdi-klzW0UZ?usp=sharing

Using Mathematica for ODEs (Ordinary Differential Equations) Playlist: https://www.youtube.com/playlist?list=PLmU0FIlJY-MlfXdrINHtYvBJINVIfQu7z

Visual Linear Algebra Online (at infinityisreallybig.com): https://infinityisreallybig.com/category/linear-algebra/

Infinite Powers, How Calculus Reveals the Secrets of the Universe (by Steven Strogatz): https://amzn.to/2XXRCF6

Another Differential Equations lecture I made: Differential Equations: As Much As You Can Possibly Learn About in 50 Minutes, especially Population Models: https://www.youtube.com/watch?v=thX3bgmRbkc

Check out my blog: https://infinityisreallybig.com/

Bethel University is a Christian liberal arts university in St. Paul, Minnesota with strong science, engineering, mathematics and computer science departments. You can also get to know your professors personally. https://www.bethel.edu/

(0:00) Last lecture about Laplace Transforms
(1:53) Mathematica solution to discontinuous sinusoidal forcing Laplace transform example
(8:52) Impulse forcing
(10:11) A first approximation to the external forcing function
(13:04) Laplace transform of the approximation
(17:46) Laplace transform of “limiting function”
(18:40) The Dirac Delta "function"
(21:10) Laplace transform of the Dirac Delta function
(22:07) This can all be made rigorous using “generalized functions” from the Theory of Distributions (linear functionals)
(24:33) Example: solve y’’+2y’+5y = δ3(t), y(0)=1, y’(0)=1
(28:07) Graph the solution, noting the lack of differentiability in the position at t = 3 (and the discontinuity in the velocity)

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