The DE Tutorial is currently being split up into different threads to make editing these posts easier.
Laplace Transforms (Part I - Introduction, IVPs and Partial Fraction Techniques)
There are many types of transformations out there. For example, differentiation and integration are types of linear transformations. However, there is one particular transform that we would like to analyze. This transform is of the form:
where is called the kernel of the transformation.
In this case, we are interested in the transform with a kernel of . With this kernel, we take and transform it into another function . This transformation described by is called the Laplace Transform. It is denoted by .
Before we go and derive all the common Laplace Transforms (we will derive many more as we get futher into later posts), let us take a look at a familar function to some of us (this may also be totally knew to some of you out there).
Given , where , we define the Gamma Function . It has the property and .
Now, if , then it follows by a similar idea that . If we continue simplifying, we have
This implies that when , .
(Thus it is interesting to point out that since , an identity for factorials.)
This will conclude the first post on Laplace Transforms. I'm not sure when I will be able to post again, now that I start classes today. I'll try to find some time in the next several weeks to do so.
Feb 8th 2011, 11:56 PM
Chris L T521
What does the future hold for the Differential Equations Tutorial?
To the MHF Community,
It's been over a year since I have updated this differential equations tutorial of mine.
Some of you have PMed me about errors/typos found in this tutorial, why certain topics aren't covered, etc. But I just haven't had the time lately to fix them or add missing topics since most of my time is devoted to my coursework (since I'm a full time Ph.D. student now). However, this summer (starting in June), I will be redoing this entire differential equations tutorial and re-release it as a mini-book (it may not be mini when I'm through with it... (Rofl) ) of some kind; meaning, it will have the theory explained in a more, precise detail. It will also have examples illustrating various techniques outlined in each section as well as a few exercises for you to attempt. At first I will not include solutions to these exercises, but in the months after releasing this to the public on MHF I will include solutions to selected exercises outlined in each section/chapter (the format is still yet to be decided).
The chapter outline will be something like:
Chapter 0 - Calculus Review
Part 1 - Ordinary Differential Equations (to be released end of summer 2011)
Chapter 1 - First Order Differential Equations Chapter 2 - Second and Higher Order Differential Equations Chapter 3 - Solving Differential Equations by Numerical Methods Chapter 4 - Matrix Methods and Systems of Differential Equations Chapter 5 - Laplace Transforms Chapter 6 - Power Series Solutions to Differential Equations
Part 2 - Partial Differential Equations (to be released sometime in 2012)
Chapter 7 - Introduction to Partial Differential Equations Chapter 8 - Fourier Series Chapter 9 - The Wave, Heat, and Laplace Equations Chapter 10 - Solving Partial Differential Equations by Various Methods Chapter 11 - Green's Functions Chapter 12 - Solving Partial Differential Equations by Numerical Methods
If you would like to contribute to this project, send me a PM and we can discuss it in more detail. If I use any of your work in my book, you will be acknowledged. :)
I will close this thread in the meantime, and I will post updates when I'm able.