What differential equations need to be solved more quickly?

[want2graduate]

For what differential equations would having much quicker or financially cheaper methods of solving them significantly benefit scientists or engineers?

 

[berkeman]

For what differential equations would having much quicker or financially cheaper methods of solving them significantly benefit scientists or engineers?

That's an *extremely* broad and ill-defined question. Could you define what you mean by "quicker" and "financially cheaper"? Are you asking about DEs or systems of DEs that require computational solutions in industry? Can you explain what you mean by "benefit" to scientists and engineers?

 

[want2graduate]

I'll rephrase.

Let's say one wanted to do original research as a university student on algorithms to solve certain differential equations or systems of differential equations. What (individual or systems of) differential equations would be worth focusing on?

By "benefit", I mean that they would have some reason to use my new method (if I were to come up with one) rather than what the currently use to solve the (systems of) difeqs. But I'm also interested in methods to solve (systems of) difeqs that are only of theoretical interest as well.

 

[berkeman]

I'll rephrase.

Let's say one wanted to do original research as a university student on algorithms to solve certain differential equations or systems of differential equations. What (individual or systems of) differential equations would be worth focusing on?

By "benefit", I mean that they would have some reason to use my new method (if I were to come up with one) rather than what the currently use to solve the (systems of) difeqs. But I'm also interested in methods to solve (systems of) difeqs that are only of theoretical interest as well.

I think it would be a better approach to look for complex problems that could benefit from better approaches to their simulation and computational solution.

Like, look for important problems in aerodynamics (high mach number airfoils, scramjet optimization, etc.), and fluid dynamics (stealthy ships and subs, etc.), optimizing wind farms with adaptive airfoils, and so on. Do you routinely read scientific journals that deal with computational modeling and applications? What are some of the more interesting applications that you've read about?

 

[berkeman]

BTW, what level are you at university? What are you graduating from, and do you have plans for more school after you graduate? Or are you headed to industry?

 

[want2graduate]

This is for my Bachelor's degree in computer science (last semester as an undergraduate). I have to do a senior thesis, and I got thinking about alternative models of computation (like analog, massively parallel, biologically inspired, ...) and was curious about figuring a cool way to solve a certain type of difeq problem (the focus would be on the alternative computational model part, and I just thought difeqs could possibly be something an alternative computational model could excel at). I'm only going to be working on this thesis until May 2017, and I'll probably join the work force afterwards. I just wanted to see if there's any possible new perspective I could probe and make some original progress on.

I'll look into your suggestions. I admittedly don't regularly read any scientific journals. I mostly just try to generally keep up to date with what's going on in computer science in general.

 

[berkeman]

I admittedly don't regularly read any scientific journals

Give that a try to help with ideas for your project. The journals should be available through your university library.

 

[aheight]

I'll rephrase.

Let's say one wanted to do original research as a university student on algorithms to solve certain differential equations or systems of differential equations. What (individual or systems of) differential equations would be worth focusing on?

By "benefit", I mean that they would have some reason to use my new method (if I were to come up with one) rather than what the currently use to solve the (systems of) difeqs. But I'm also interested in methods to solve (systems of) difeqs that are only of theoretical interest as well.

Without a doubt, non-linear ones. First though, have a good background on ordinary DEs.