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/