applications of laplace transforms
DESCRIPTION
Laplace transforms have huge number of application in computer system. Let's have a look upon some of the applications. Please find folder C4 from the following link http://goo.gl/bNwxMbTRANSCRIPT
Context 1. Introduction to Laplace Transforms.
2. Why Laplace?
3. Applications in daily life.
4. Practical analysis of change of scale property.
5. Overview to applications of Laplace in various engineering fields.
6. Limitations of Laplace Transform.
7. Conclusion of the Assignment.
*Reference Sources.
1. Introduction to Laplace Transforms
Introduction: The Laplace Transform was first used by a French mathematician and astronomer
named after Pierre Simon Laplace. He develop the Laplace operator which have large number of
applications in field of physics and Astroengineering.
Definition: If f(t) is a function for all positive values of t then Laplace of f(t) will be:
[ ] ∫
Where F(S) is known as Laplace transform with the above integral exists.
Practical meaning: Basically, Laplace transforms are used to change domain of the function
without change it’s original value. Think that one person says 2 by 2 = 4 and another says 3+1=4
here, the value of function is same but domain is different. That’s same the Laplace does in the
field of science. Let’s take an other example:
[ ]
If we observe the above example, here we just change the function domain from time to
frequency but the value of function does same. This all just because of behavior of s domain of
Laplace transform.
2. Why Laplace?
Well we have so many transforms so, a question usually come to our mind that why to use
Laplace. There is a simple answer that Laplace transform have approach to conservative law of
nature. When we apply Laplace to a function then nothing is gained and nothing is lost but the
form of function has been changed.
Laplace transforms are very useful to convert complex differential equation into relatively simple
equations. So, the equations having polynomials are easier to solve, we use Laplace transform to
make calculations easier. The complex differential equation can be simplified with the help of
following equation:
[ ] [ ]
3. Applications in daily life.
We are living in the age of technology and the
technology we mostly used is mobile phones or cell
phones. Laplace transforms are also used in
modulation of signal. Whenever the signal have been
transmitted it has been convert into frequency
domain and when the receiver receives it, again
convert into time domain. The main idea is that the
matter is revolve about Laplace transform of function
and Laplace inverse transformation of function.
This technology almost has been used in all 2-way
wireless communication.
Image credit: http://3.imimg.com/data3/XK/NF/MY-
4231085/mobile-tower-500x500.jpg
4. Practical analysis of change of scale property.
Definition: If [ ] then:
[ ]
(
)
Practical meaning: If we multiply any constant to value of t then we will get the above result
but the overall value of function will remain same.
Demonstration: Open the folder C4 and listen the music of “file 1.mp3” and after listen it open
the music of “file 2.mp3” and also listen it. You will found the length and music of the both files
is same but the voice of singer is a different. Actually I apply stretch to the audio of file 2 in
Adobe Audition CS6 which follows the algorithmic approach to Laplace transform to change the
pitch of voice and the visual representation of the both files will be same as shown below.
Visual graphical representation of audio of file 1 and file 2
5. Overview to applications of Laplace in various engineering fields.
Due to simple method of transformations. The Laplace transforms has been used in various
engineering fields such as:
1. System Modeling: In system modeling we have to deal with large number of differential
equations and we use Laplace transforms to simplify these equations.
2. Analysis of Electrical Circuits: Laplace transforms are also use to analyze time in variant
electrical circuits.
3. Analysis of Electronic Circuits: Laplace transforms are also used by electronics engineer to
solve linear equations and it is also use in MATLAB.
4. Digital Signal Processing: As we discuss in last page. We actually processed digital signal
with the help of Laplace. We even can’t imagine digital signal processing without the Laplace
transformation.
5. Nuclear Physics: In nuclear physics, Laplace transform is used to get the true form of
radioactive decay. It make us possible to studying analytic part of Nuclear Physics.
6. Process Controls: Laplace transforms is widely use in process controls. It use to analyze the
variables, when the value has been change, produces desired manipulations in the result. In the
study of heat experiments Laplace transform is used to find out, what extent the given input can
be altered by changing temperature, hence one can alter temperature to get desired output for a
while. This is an efficient and easier way to control processes that are guided by differential
equations.
6. Limitations of Laplace Transform
1. Laplace transform is just another way of solving differential equations. It is used because it
makes calculations much easier, hence there are some limitations to it.
2. An Engineer won’t be able to use Laplace transform to solve anything but a linear ODE
(Ordinary Differential Equation) and that too with constant coefficients. Laplace transforms can
be used to solve linear ODEs where coefficients are not known by first finding a substitute value,
which converts the given DE into a DE with constant coefficients.
7. Conclusion of the report.
As we study that Laplace is base of digital signal
processing and nuclear physics. Without Laplace
the respective branches are meaningless and
instead of these it has wide level applications in
electrical and electronics and also have a major
role in branch of physics. Laplace transform is
time efficient, this is also a reason that it is
widely adopted.
Image Credit: Manipulated in Photoshop CS6
Reference Sources:
Books:
Higher Engineering Mathematics by H.K Dass, Er. Rajnish Verma.
Links:
http://jdebug.org/practical-applications-of-laplace-transform/
http://en.wikipedia.org/wiki/Laplace_transform
http://en.wikipedia.org/wiki/Astroengineering
https://www.khanacademy.org/math/differential-equations/laplace-transform/laplace-transform-tutorial/v/laplace-transform-1
http://ocw.mit.edu/courses/mathematics/18-03sc-differential-equations-fall-2011/unit-iii-fourier-series-and-laplace-transform/laplace-transform-basics/