Problems have been, as a whole, classified, thanks to Computational Complexity. But, in differential equations, is it possible to classify differential equations depending on their computational structure?
For example, if a first order non-homogeneous equation is comparatively difficult to solve than a, say, 100th order homogeneous equation, can they be classified as separate convexity classes, given the method to solve was same? If we vary the process of solving, how random shall the solutions, their existence and stability, and other properties vary?
I would assume that I am partly convinced that solving differential equations might be NP-Hard:
http://mathoverflow.net/questions/158068/simple-example-of-why-differential-equations-can-be-np-hard
This chapter:
http://link.springer.com/chapter/10.1007/978-1-4757-2793-7_20
this article:
http://www.cs.princeton.edu/~ken/MCS86.pdf
and this presentation:
http://www-imai.is.s.u-tokyo.ac.jp/~kawamura/slides_cambridge.pdf
have been forcing me ask for the scope of computational complexity according to the solvabality of Differential Equations. Starting with ordinary differential equations, we could classify partial, delay, difference equations etc.
I had once thought of incorporating dynamic programming using the iterates that were calculated while approximating a solution, but lost myself somewhere.
This is a good question with more than one possible answer.
The analog characterization of a complexity class for linear differential equations is given in
J.F. Costa, J. Mycka, What lies beyond the mountains, 2004:
http://www.mimuw.edu.pl/~salwicki/PPIM/mountains.pdf
This is a good question with more than one possible answer.
The analog characterization of a complexity class for linear differential equations is given in
J.F. Costa, J. Mycka, What lies beyond the mountains, 2004:
http://www.mimuw.edu.pl/~salwicki/PPIM/mountains.pdf
x(t) = f(x(t))
There is no general method for solving this system, hence the most important way of getting information about the solutions is to determine the phase portrait. The constant solutions x(t) ≡ p can be obtained by solving the system of algebraic equations f(p) = 0. The solution p of this system is called an equilibrium or steady state of the dynamical system.
http://www.cs.elte.hu/~simonp/dynsysdiffeq.pdf
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