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 November 10th, 2018, 10:12 AM #1 Newbie   Joined: Nov 2014 From: Deutschland Posts: 2 Thanks: 0 Transformation of Equations - What is the deeper thing? Hi, If you have an ordinary differential equation (or equations), you can transform them under some conditions (integral, linear operator, ...) to algebraic equations. If you have a partial differential equation you may transform it to an ODE. This can be shown "easily" by doing Fourier or Laplace transforms. My fundamental question is: What "mathematical mechanism" is used here? Is there a deeper fundamental level hidden? I hope you know what I mean, thanks! Best, Sin November 14th, 2018, 09:43 AM   #2
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 Originally Posted by Sin Hi, If you have an ordinary differential equation (or equations), you can transform them under some conditions (integral, linear operator, ...) to algebraic equations. If you have a partial differential equation you may transform it to an ODE. This can be shown "easily" by doing Fourier or Laplace transforms. My fundamental question is: What "mathematical mechanism" is used here? Is there a deeper fundamental level hidden? I hope you know what I mean, thanks! Best, Sin
Yes there is an underlying reason.

Lazyness

The transformed equation(s) are easier to solve than the original, at the expense of having to transform back at the end.

The simplest such transformation is the logarithmic one which was used for a long time to make multiplication/division and exponentiation of numbers easier.
That is the problem of multiplication was transformed to one of addition (of logs) at the expense of having to find the anti-log at the end.

That was the basis of the old fashioned slide rule November 14th, 2018, 11:52 AM #3 Senior Member   Joined: Feb 2016 From: Australia Posts: 1,838 Thanks: 653 Math Focus: Yet to find out. Maybe not a more 'fundamental', but a more general idea is that of the integral transform: https://en.wikipedia.org/wiki/Integral_transform Thanks from topsquark Tags deeper, equations, fourier, laplace, meaning, ode, pde, thing, transformation Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post alan2here Math 6 August 9th, 2018 03:06 PM skipjack Math 1 July 16th, 2018 04:41 AM Antoniomathgini Calculus 5 December 6th, 2017 08:41 PM

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