intuition about definition of laplace transform


by learner07
Tags: definition, intuition, laplace, transform
learner07
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#1
Nov28-12, 05:32 AM
P: 6
why was laplace transform developed i have googled it and found that it is something about shaping a family of exponential and vector projections etc i couldn't get it. some simply said that it was used to make a linear differential equation to algebraic equation but i couldn't understand how the variable t(time ) went in and how the 's' variable popped out. could you guys please explain me about how this laplace transform actually works? and when u say L(1)=1/s what does that actually mean ?

thanks in advance
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okami11408
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#2
Nov28-12, 05:43 AM
P: 15
The simplest way of understand Laplace transform is to understand Fourier transform.

Laplace transform is Fourier transform of exponential decay multiply by the given function.

This happened because we cannot Fourier transform some functions (the value goes to infinity) therefore we multiply it with an exponential decay first, then transform it,

this method is called "Laplace Transform"

Hope this help
okami11408
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#3
Nov28-12, 05:49 AM
P: 15
when you say L(1)=1/s, it means that 1 in time domain is 1/s in Laplace domain.

Laplace domain is frequency domain of a function that already multiply with exponential decay.

dkotschessaa
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#4
Nov28-12, 07:02 AM
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intuition about definition of laplace transform


Is it common to learn Laplace transformations without learning Fourier? We did not learn Fourier at all and my intuition is sketchy as well.
yungman
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#5
Nov28-12, 11:38 AM
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The "s" is a definition the transform of a function:

[tex]L(f(t))=f(s)=\int^{\infty}_0 e^{-st} f(t)dt[/tex]

There is no explanation on this, it is a definition.

The use of it is to solve an equation that has integration and differentiation, into a simple algebraic equation.
yungman
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#6
Nov28-12, 11:53 AM
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Quote Quote by dkotschessaa View Post
Is it common to learn Laplace transformations without learning Fourier? We did not learn Fourier at all and my intuition is sketchy as well.
I don't particularly see you need to learn one in order to learn the other. They are similar in the sense of the definition and the formula of the transform, but that's about it. You can definitely learn Laplace without learning Fourier.

But again like in the other post, enroll in the ODE and PDE class, they teach you all these. I cannot emphasize how much I don't like math, but it's a necessary evil in engineering. Learn both of them.
dkotschessaa
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#7
Nov28-12, 12:17 PM
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Funny, I actually *like* math (math major) but don't like Differential equations because it seems so "engineering-y." lol

In my DE class we learned Laplace but not Fourier - going onto series now.

-Dave K
yungman
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#8
Nov28-12, 12:39 PM
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Quote Quote by dkotschessaa View Post
Funny, I actually *like* math (math major) but don't like Differential equations because it seems so "engineering-y." lol

In my DE class we learned Laplace but not Fourier - going onto series now.

-Dave K
Series is actually very important as a lot of problems cannot be solved by conventional means and resort to numerical approx. by power series etc. Case in point, Bessel and Lagendre function are good examples of this.
Runei
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#9
Nov28-12, 01:05 PM
P: 135
Hello there

I have had the exact same thoughts as you, and I believe I have found out how you can learn more, but - I must warn you - it is not an easy way.

The best way to do it is to learn about Linear Transformations in Linear Algebra. Linear Algebra and differential equations can be coupled together quite nicely. The thing is that we want reversible Linear Transformations where the "information" is not lost when we do the transformation.

It has to do with change of basis and eigenvalues. Using these concepts its possible to get a quite intuitive idea of the laplace transform.

Fourier Transform is a simple case of decomposition, while the Laplace transform is not like that. It has a more analytical idea, that stems from the concept of a Linear Transformation.

So if you're willing. Learn Linear Algebra and then dig into Linear Transformations. Learn about how to couple Linear Algebra and differential equations, and then the Laplace transformation will be a linear transformation of a "vector" that describes the differential equation.
dkotschessaa
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#10
Nov28-12, 01:05 PM
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Quote Quote by yungman View Post
Series is actually very important as a lot of problems cannot be solved by conventional means and resort to numerical approx. by power series etc. Case in point, Bessel and Lagendre function are good examples of this.
Yeah, series I find interesting, if sometimes elusive.
yungman
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#11
Nov28-12, 02:26 PM
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Quote Quote by dkotschessaa View Post
Yeah, series I find interesting, if sometimes elusive.
Ha ha!!! I don't. It's just necessary evil!!! Look into Bessel and Lagendre, you'll love it. It is very important for boundary condition problem in EM when using cylindrical and spherical coordinates.
learner07
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#12
Nov29-12, 08:39 AM
P: 6
hi to all i don't need math definitions.what i need is physical understanding/interpretations of what laplace transform actually is ? and how does it work every text book gives me definitions (for those how have posted me regular definitions) @runei please tell me more about what u discussed above i mean how should i start etc etc and are their any other ways also if there please mention?
jim hardy
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#13
Nov29-12, 10:37 AM
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I have hoped for years to find a physical explanation or analogy for Laplace.

To me the Laplace transform is a tool that turns complex equations into simple ones.

One needn't understand impulse and momentum and fracture mechanics to use a hammer..
but he can become adept at its use through repetition.

Perhaps someday that AHA! moment will come.

I can only envy those who do understand. I accept my limitations, and appreciate those who share their deeper understanding.

old jim
dkotschessaa
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#14
Nov29-12, 10:44 AM
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Perhaps this is a case of.. "Young man, in mathematics you don't understand things. You just get used to them..." (John Von Neumann)
yungman
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#15
Nov29-12, 11:15 AM
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Quote Quote by learner07 View Post
hi to all i don't need math definitions.what i need is physical understanding/interpretations of what laplace transform actually is ? and how does it work every text book gives me definitions (for those how have posted me regular definitions) @runei please tell me more about what u discussed above i mean how should i start etc etc and are their any other ways also if there please mention?
Read my last post, it is a definition. It is like I am making up my transform called Alan transform and is defined as

A(f(t))= f(t)+1

It is absolutely useless, BUT it's a transform!!!! Difference is my transform don't worry anything and Laplace transform works!!!

Their might not be any rhyme and reason at all. There are a lot of things like this in science and math. The higher level you go, the more you encounter, try Fourier transform, Bessel, Legendre, try to make sense where it came from. It a big miss conception that science comes from theory, then proved by observation. A lot of them started from observation, then make up the theory and math. Some by intuition. Do you know how they discover the benzene ring in chemistry? The guy had a dream of a snake biting it's own tail and it goes around and around!!! Then he went on and proved that was true!!!! That's part of the reason I quit chemistry after I got the degree, I thought everything has a reason and from theory to practice...........I was wrong and I want to have nothing to do with it.

You are in the wrong section here, go to "Differential Equation" in the math section here and ask. This is math.
Runei
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#16
Nov29-12, 12:41 PM
P: 135
I can try and tell you how I got my kind of intuition first.
  1. Learn Linear Algebra. Look at Khan Academy, and I can recommend the book "Linear Algebra and it's Applications 4th" by David C. Lay.
  2. Pay special attention to the Linear Transformations places and do some of the problems
  3. Learn that FUNCTIONS ARE VECTORS!!! This is an important part
  4. Learn how to couple Linear Algebra and Differential Equations
  5. Learn how the Laplace Transform is a Linear transformation

I can also recommend the following websites where they talk about it:

1. http://www.quora.com/Intuitively-spe...tion-represent (the first answer is a guy explaining exactly what I am talking about, but if you havent learned Linear Algebra, it is hard to understand)

2. http://mathoverflow.net/questions/28...ral-transforms (look at the answer from John D. Cook - answer number 3 i believe. He gives a possible geometric interpretation)

3. http://www.youtube.com/watch?v=sZ2qulI6GEk (his introduction has another good interpretation)

The thing is - There can be MANY interpretations of the transforms. But learning alot of them will give you more and more of the puzzle. At some point. You will get it.
Ratch
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#17
Nov29-12, 02:46 PM
P: 315
l7,

why was laplace transform developed
Why were logarithms developed? So one could multiply and divide by adding and subtracting.

Why is the Laplace transformed? So one could manipulate and solve differential equations by algebraic means without calculus.

but i couldn't understand how the variable t(time ) went in and how the 's' variable popped out.
Did you ever try to integrate a simple function using the Laplace integral to observe how the "e" term disappears when the upper limit "t" goes to infinity. And how it becomes one when the lower limit is zero?

could you guys please explain me about how this laplace transform actually works? and when u say L(1)=1/s what does that actually mean ?
There are a hundred thousand textbooks that do that. Why should any of us duplicate that effort over again? 1/s is the Laplace transform the the unit step function. That is explained many times in textbooks also.

Ratch

P.S. In English, sentence beginnings are capitalized. It makes them easier to read that way.
AlephZero
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#18
Nov29-12, 03:25 PM
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Instead of getting tied in knots about how the transform works, it might be better to learn what the s-plane represents - for example what the locations of poles and zeros mean for the dynamic behaviour of a system.


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