We use transformations to convert the PDE into ODE to find the exact solutions of the PDE. If we use the different definitions of derivative in the transformations, then is it possible to obtain a different ODE of the given same PDE?
Possibly-but that doesn’t matter, since the solution of the PDE is the same.
The answer is a big NO, and in order to understand you need to know the fundamental theorem of existance and uniqueness of the ODE systems.
see e.g., https://www.youtube.com/watch?v=_WpncZ3RkTg
Sorry, I don't get it. What do you mean by the exact solution for the PDE?
As far as I know, we transform a PDE (P) into particular ODEs (Pn) using, for example, the definition of a derivative to approximate the PDE by the ODEs. At this point, we can find solutions for ODEs that serve for constructing approximate solutions for the PDE. And the hope next is to prove that these approximate solutions converge somehow to a type of solution of the PDE.
Returning to your question, I don't see why not. From the description above, it is like saying that is it possible to find different sequences (real) that converge to the same number (real)!!?
Have a good day :)
What type of PDE are you going to transform? And what transformation are you using?
First of all not all PDE's have solutions - even locally.
https://en.wikipedia.org/wiki/Lewy%27s_example
The trick of generating ODE's from PDE's requires that the variables can be separated. That is only true in a limited case. https://math.stackexchange.com/questions/863740/when-is-separation-of-variables-an-acceptable-assumption-to-solve-a-pde
The first thing to consider when addressing any P.D.E is existence and uniqueness of solution. This is always reliant on the boundary value constraints imposed (you can read my most recent write-up for more pointers, perhaps). Without imposition of any boundary value constraints, you can probe reducibility of PDE's to ODE's, which depends on admissibility of point symmetries. If an equation admits more than one group of Lie symmetries, it may be reducible to more than one ODE. This is when the boundary value constraints need to be reckoned with, and the essence/aesthetics of PDE's interface with Symmetry Theory and Trace Theory can be brought to the fore. Anyone else with foresight into the natural analytic solution techniques and their scientific interpretations via conservation laws should kindly contact me: I'm trying to learn more on this.
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