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want2graduate

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In summary, the conversation discusses the possibility of finding quicker or financially cheaper methods to solve differential equations, specifically for the benefit of scientists and engineers. The speaker suggests focusing on complex problems in aerodynamics and fluid dynamics, and recommends reading scientific journals for ideas. They also mention their interest in alternative computational models and solving non-linear differential equations.

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want2graduate

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berkeman

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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 said:

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want2graduate

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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.

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berkeman

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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.want2graduate said:

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.

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?

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berkeman

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want2graduate

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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.

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berkeman

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Give that a try to help with ideas for your project. The journals should be available through your university library.want2graduate said:I admittedly don't regularly read any scientific journals

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aheight

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want2graduate said:

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.

Differential equations are mathematical equations that describe how a quantity changes over time, based on the rate of change of that quantity and its current value.

Differential equations are used to model real-world processes in various fields, such as physics, engineering, and economics. Being able to solve them quickly allows for faster and more accurate predictions and analysis of these processes.

There are various methods for solving differential equations, including analytical methods (e.g. separation of variables, variation of parameters) and numerical methods (e.g. Euler's method, Runge-Kutta methods). The most appropriate method depends on the type and complexity of the differential equation.

Yes, technology such as computers and specialized software can be used to solve differential equations quickly. These tools can handle complex equations and perform calculations much faster than manual methods.

Yes, there are some challenges in solving differential equations quickly. Some equations may be too complex for current technology to handle, and certain methods may not be suitable for certain types of equations. Additionally, a high level of accuracy may be required, which can also affect the speed of solving differential equations.

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