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Differential equations and integal transforms

  1. Feb 24, 2007 #1
    Differential equations and integral transforms


    I have some general questions on using integral transforms for solving differential equations.

    Also, I know that Fourier and Laplace transforms are useful means for solving linear ODE's and PDE's.
    1. Are there cases, when one of them is more powerful as another one?
    2. What about Hankel and Mellin transforms? Are there also very powerful?
    3. I think, that Laplace transforms can be applied only for Cauchy problems (initial conditions must be given). Is it correct?

    Thank you in advance
    Last edited: Feb 24, 2007
  2. jcsd
  3. Feb 25, 2007 #2
    Should i ask such questions in Homework Forum?! :confused:
    It's not a homework at all... just hoping that someone have more experience with such things...
  4. Feb 25, 2007 #3
    It is a pretty broad question. To answer this would be to place one's self out on a limb and speculate. But, here is an attempt.

    The last two types you mentioned are sort of specialized. Hankel transforms are related to Fourier transforms in that they are FT of radially symmetric functions. Mellin are even more specialized and have some questions of convergence. (Some well-known functions can be viewed as a Mellin transform.)

    Very roughly though (and I am sure there are cases where some may disagree) FT and LT are most common: The one you choose would depend on what sort of kernel (exponential) you want. The kernel for the FT has modulus 1 and the kernel for the LT has exponential decay. The exponential decay of the LT is nice since you can transform some functions that do not have a FT and you can do the so-called Heaviside calculus (procedure which puts the initial conditions into the problem and transforms the diffeq to an algebraic equation). Try looking all these up on Mathworld

    So, there is an answer in a nutshell, but the question is bigger than a watermelon.
  5. Feb 26, 2007 #4
    Thank you very much for interesting information!!
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