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Intuition about definition of laplace transform

  1. Nov 28, 2012 #1
    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
     
  2. jcsd
  3. Nov 28, 2012 #2
    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
     
    Last edited: Nov 28, 2012
  4. Nov 28, 2012 #3
    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.
     
    Last edited: Nov 28, 2012
  5. Nov 28, 2012 #4
    Is it common to learn Laplace transformations without learning Fourier? We did not learn Fourier at all and my intuition is sketchy as well.
     
  6. Nov 28, 2012 #5
    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.
     
    Last edited: Nov 28, 2012
  7. Nov 28, 2012 #6
    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.
     
  8. Nov 28, 2012 #7
    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
     
  9. Nov 28, 2012 #8
    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.
     
  10. Nov 28, 2012 #9
    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.
     
  11. Nov 28, 2012 #10
    Yeah, series I find interesting, if sometimes elusive.
     
  12. Nov 28, 2012 #11
    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.
     
  13. Nov 29, 2012 #12
    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?
     
  14. Nov 29, 2012 #13

    jim hardy

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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
     
  15. Nov 29, 2012 #14
    Perhaps this is a case of.. "Young man, in mathematics you don't understand things. You just get used to them..." (John Von Neumann)
     
  16. Nov 29, 2012 #15
    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!!!!:rofl: 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.
     
    Last edited: Nov 29, 2012
  17. Nov 29, 2012 #16
    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-speaking-what-does-a-Laplace-transformation-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/2809/intuition-for-integral-transforms (look at the answer from John D. Cook - answer number 3 i believe. He gives a possible geometric interpretation)

    3. (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.
     
    Last edited by a moderator: Sep 25, 2014
  18. Nov 29, 2012 #17
    l7,

    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.

    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?

    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.
     
  19. Nov 29, 2012 #18

    AlephZero

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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.
     
  20. Nov 29, 2012 #19
    Laplace Transform is one of the simpler ones that has a calculus representation already. In real world, quite a bit of result can only be represented by numerical analysis which is some form of power series. Try to make sense out of that!!! People might be plotting out an input to output response and literally doing a curve fitting to develop an equation for it.

    Like Ratch gave, log is one of those, why do log? From my understanding, one reason is human ears response logarithmic to sound power, there must be a lot of other reason, but that's good enough for me!!!

    I remember the first lecture from the instructor of my Physical Chemistry that really stuck in my mind: Invention or discovery of a theory start with an idea or postulation. Then confirm or disprove using observation and experiment. If the postulation holds after scrutiny, then it is consider true and become a law. At the same time, mathematical formulas are developed to full fill the observation. Sometimes the postulation is proven by mathematical derivation. It is not the other way around that you have a theory or equation first.

    If you get stuck with Laplace, try quantum physics!!! I quit chemistry after I score the first in the class of the Physical chemistry and I was like 15 points above the second student. I worked in the chemistry dept. at the time and it's like the professor came to the stockroom window everyday answering my questions. Finally he said to me " Alan, you are not going to understand this, keep at it, when you get your PHD, you'll start to get the feel of it". This might not be the exact word to word, but it's very close. I quite chemistry after that, I just finished my degree and never even look for a job in that field. Between the snake biting it's tail and this, I quit. I did not know at the time it's like this in all science. Now I can really appreciate his first lecture and what he said.
     
  21. Dec 1, 2012 #20
    Functions are vectors and therefore they can be represented in any coordinate system you want. Just like when you apply a transformation on a spatial vector to transform from xyz coordinates to polar, the Laplace transforms the function by projecting it onto a different space.

    I tried to think of an analogy that would make this idea a little more tangible and I came up with a technical drawing. A technical drawing uses three projections to make a 3D object seem simpler to the eye. You could choose to represent a 3D object with an animated model that rotates in time, instead. If you did that, it would be very easy for your eye to interpret the picture and get a good impression of the object's shape, but measuring precise lengths and angles would be very hard, and you would have to have some kind of magic paper to display it. It becomes simpler to represent it with still images in three different locations on a piece of paper. In these two different representations, the viewing angle is either a function of time or location, and the relative length of a line may or may not depend on relative distance. We transform the coordinates to a equivalent but more easily understood system. We also choose to align the axis of the drawing in a way that is convenient for design and construction, the same object could be represented with a drawing from three other arbitrarily chosen orthoganal directions but that would be confusing to your eye.

    I think this is similar to the situation of the laplace transform because its useful for transforming time-varying functions into inanimate pictures in which time dependence is represented by spatial coordinates. In S-space, periodicity becomes a length along one axis, and the slope of exponential decay becomes a length along the other axis, any functions of S-space that are multiplied together represents the convolution of their time-varying counterparts (which is why its useful for filter analysis), and phase angles look like... angles. Like in the technical drawing example, drawing things in different locations now represents something totally different and makes it easier to express certain ideas.

    S-space is a nice place to work if your job is doing a lot of convolution, differential equations, and phase analysis. The laplace transform is like the train that you take to commute from regular space where you live to S-space where you work.

    Once you've mastered the Laplace Transform, try the fractional Fourier Transform... I still don't get that one
     
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