Current DE Research: Math Physics & Unsolved Areas

[Parmenides]

Hello Everybody,

I currently study physics and math as an undergraduate and the area of differential equations is of great interest to me (despite being immensely challenging!). I wanted to peer into the current development of differential equations in mathematical physics and if there remains any modern areas of research that remain unsolved or of interest to the professional/academic world. Perhaps there are some previous pages that discuss this that somebody could refer me to? Thank you!

 

[bigfooted]

What would you like to do exactly? Study model equations of physical problems, or work on mathematical proofs of properties of differential equations?

which field of mathematics/physics are you interested in? I guess every field has its own specific problems.

If you want to focus on problems with finding solutions of differential equations, independent on any physics, then a problem like finding the canonical form of pde's (finding the differential Groebner basis) is interesting to investigate. Even a seemingly simple thing like a closed-form algorithm for general first order ode's does not exist yet.

 

[diligence]

Even a seemingly simple thing like a closed-form algorithm for general first order ode's does not exist yet.

What exactly do you mean by closed-form? Do you mean an algorithm for any 1st order ODE in general?

 

[bigfooted]

What exactly do you mean by closed-form? Do you mean an algorithm for any 1st order ODE in general?

I mean an algorithm that is closed in the sense that you do not need an ansatz (an initial guess) to solve a 1st order ODE. Current solution methods in e.g. Maple go through a list of common ansatze and check if the ODE can be solved by them. For instance, they check if the ODE is translational invariant. Once we know an additional property of the ODE we can solve it, but we don't know how to get the additional property in a systematic way.