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November 10th, 2018, 11:12 AM   #1
Sin
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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!

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Sin
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November 14th, 2018, 10:43 AM   #2
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Quote:
Originally Posted by Sin View Post
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
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November 14th, 2018, 12:52 PM   #3
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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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