
Differential Equations Ordinary and Partial Differential Equations Math Forum 
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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  
Senior Member Joined: Jun 2015 From: England Posts: 915 Thanks: 271  Quote:
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 antilog 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: 1,834 Thanks: 650 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 

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deeper, equations, fourier, laplace, meaning, ode, pde, thing, transformation 
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