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.

 

[blue_raver22]

I am pleased to see your interest in the field of differential equations and its applications in mathematical physics. This area of research has a long history and continues to be a vital part of many scientific fields. In fact, many of the fundamental laws and principles in physics, such as Newton's laws of motion and Maxwell's equations, are described using differential equations.

In terms of current research, there are many exciting developments happening in the field of differential equations in mathematical physics. One area of interest is the study of nonlinear differential equations, which are more complex and difficult to solve compared to linear equations. These equations have applications in fields such as chaos theory, fluid dynamics, and quantum mechanics.

Another area of active research is the use of differential equations in understanding and modeling complex systems, such as biological systems and climate change. These systems involve multiple variables and interactions, making them challenging to analyze and predict. Differential equations provide a powerful tool for studying these systems and making predictions about their behavior.

As for unsolved areas, there are still many open problems in the field of differential equations in mathematical physics. For example, the Navier-Stokes equations, which describe the motion of fluids, are still unsolved for certain boundary conditions. Additionally, there is ongoing research in developing new methods for solving differential equations, as well as studying their properties and behavior.

I would recommend exploring journals and conferences in the field of mathematical physics to stay updated on the latest research and developments in differential equations. Additionally, consulting with professors and researchers in this field would be a valuable resource for learning about current research topics and potential areas of interest.

I wish you the best in your studies and hope you continue to pursue your interest in differential equations in mathematical physics. It is an ever-evolving and exciting field with endless possibilities for exploration and discovery.