Differential equations and integal transforms

[Yegor]

Differential equations and integral transforms

Hi!

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

 

[Yegor]

Should i ask such questions in Homework Forum?!
It's not a homework at all... just hoping that someone have more experience with such things...

 

[gammamcc]

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 differential equation to an algebraic equation). Try looking all these up on Mathworld
http://mathworld.wolfram.com/

So, there is an answer in a nutshell, but the question is bigger than a watermelon.

 

[Yegor]

Thank you very much for interesting information!