/sci/ - Science & Math » Thread #14218028

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Anonymous Fri 11 Feb 17:59:40 2022 No.14218028 View Reply Original Report

Quoted By: >>14218031 >>14218040 >>14218102 >>14218106 >>14218206 >>14218222 >>14219394 >>14220487

If everything can be solved numerically why bother with analytical solutions?

Anonymous

Anonymous Fri 11 Feb 2022 18:02:00 No.14218031 Report

Quoted By:

>>14218028
analytical solutions are better if you can have them, otherwise theres no reason to obsess over them.

Anonymous

Anonymous Fri 11 Feb 2022 18:04:31 No.14218040 Report

Quoted By:

>>14218028
because analytical solutions provide insights that numerical solutions just dont, like the overall behavior of a function, its derivatives or some special points of it. Numerical solutions are just values and do not have such information. When analytical solutions are not possible, numerical is good enough, but when analytics is possible, you should do analytics, because it provides a much more complete picture of the solution to your prolem

Anonymous

Anonymous Fri 11 Feb 2022 18:11:53 No.14218078 Report

Quoted By: >>14218094

Numerical solutions are not always possible. Chaotic systems often become unpredictable no matter how well you simulate them because of sensitivity to initial conditions. You can't resolve this by just computing harder with the same data.

Anonymous

Anonymous Fri 11 Feb 2022 18:15:32 No.14218094 Report

Quoted By: >>14218159 >>14218218

>>14218078
>Chaotic systems often become unpredictable no matter how well you simulate them because of sensitivity to initial conditions
None of the relevant ones have analytical solutions, either.

Anonymous

Anonymous Fri 11 Feb 2022 18:16:30 No.14218102 Report

Quoted By:

>>14218028
1. Exact solutions are exact. Numeric solutions always have error.
2. It is always good to look for exact solutions because then you can write closed form solutions for bvp/ivp.
3. There are classes of equations that are exactly integrible and there arises interesting properties in other fields like group theory and geometry.

Anonymous

Anonymous Fri 11 Feb 2022 18:17:46 No.14218106 Report

Quoted By:

>>14218028
>If everything can be solved numerically why bother with analytical solutions?
because analytical solutions are what allows us to approximate numerically :/
we employ idealistic numbers (reals, complex, etc) to simplify formulas and create shortcuts, and only at the very end we enjoy some rational numbers action

Anonymous

Anonymous Fri 11 Feb 2022 18:30:49 No.14218159 Report

Quoted By: >>14218218

>>14218094
>None of the relevant ones have analytical solutions, either.
This. No PDEs of physical significance have analytical solutions.

Anonymous

Anonymous Fri 11 Feb 2022 18:45:09 No.14218206 Report

Quoted By: >>14218223

>>14218028
because muh elegance

Anonymous

Anonymous Fri 11 Feb 2022 18:47:52 No.14218218 Report

Quoted By: >>14218534 >>14218543

>>14218159
>>14218094
Incorrect. Navier Stokes for example has a large number of analytical solutions. There an old book in French that has several analytic solutions, the main author's last name is "Bleaker" or something like that. There are also many papers that find exact solutions. That's also a similar story for Gas Dynamic Equations. L.V. Ovsyannikov has many papers on these. With Fourier Transforms all the relevant linear equations like the Heat/Wave equations have exact solutions. The linear wave equation $u_{tt} - a^2 u_{xx} =0$ has exact solutions in the nice form $u(x,t) = f(x+at) + g(x-at)$ . There are also quite a few exact solutions to Maxwell's equations, Einstein-Hilbert equations, etc. For GR, there are books that classify a thousand solutions or so of that equation alone and much of the research in physics relating to GR are based on those solutions. Often, there are equations in fluid dynamics papers that are integrible as well that the people doing the research don't know what to do to either solve them or even to determine whether or not they're integrible so they just do numerics all day.

Anonymous

Anonymous Fri 11 Feb 2022 18:49:46 No.14218222 Report

Quoted By:

>>14218028
More precision = Smaller distinctions = Bigger results

Anonymous

Anonymous Fri 11 Feb 2022 18:49:55 No.14218223 Report

Quoted By:

>>14218206
it has nothing to do with elegance. Analytic solutions are just better. Analytic solutions only have error due to the problems with rationally approximating real numbers. Numerical solutions not only share that error, but in general they introduce new error due to the fact that they're just approximations of solutions and you have to deal with error associated with rate of convergence.

Anonymous

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Anonymous Fri 11 Feb 2022 20:25:41 No.14218534 Report

Quoted By: >>14218599

>>14218218
So what exactly needs to yet still be solved for Navier Stokes? The most general form analytic solution?

Anonymous

Anonymous Fri 11 Feb 2022 20:29:12 No.14218543 Report

Quoted By: >>14218599

>>14218218
Eh? The analytical solutions are for cases that are not chaotic.

Anonymous

Anonymous Fri 11 Feb 2022 20:48:54 No.14218599 Report

Quoted By: >>14218849

>>14218543
In the case of chaotic, you really can't find a solution.
>>14218534
What needs to be solved is the global existence and uniqueness theorem for solutions
https://www.claymath.org/millennium-problems

Anonymous

Anonymous Fri 11 Feb 2022 20:55:47 No.14218621 Report

Quoted By: >>14218839

the equations you numerically solve are analytic, without analytic it just a sea of numbers

Anonymous

Anonymous Fri 11 Feb 2022 22:00:12 No.14218839 Report

Quoted By: >>14220294

>>14218621
>without analytic
Without the analytic property, you'd be opening yourself up to all sorts of things that can happen in Smooth and $C^n$ functions. Analytic is assumed more often because it means your solutions are more likely to have nicer behavior without being some exotic mess that you don't know how to work with.

Anonymous

Anonymous Fri 11 Feb 2022 22:02:39 No.14218849 Report

Quoted By: >>14219256

>>14218599
>In the case of chaotic, you really can't find a solution.
You mean in the sense that even numerical methods are basically useless? Okay, but how does that contradict my point?

Anonymous

Anonymous Sat 12 Feb 2022 00:06:56 No.14219256 Report

Quoted By: >>14220452

>>14218849
I don't know why people even mention numerical methods. They're not interesting. We get it, they work for application. But they're an approximation of little intrinsic intrigue.

Anonymous

Anonymous Sat 12 Feb 2022 00:47:57 No.14219394 Report

Quoted By:

>>14218028
god calculus has some really fucking bad notation

Anonymous

Anonymous Sat 12 Feb 2022 09:00:40 No.14220294 Report

Quoted By:

>>14218839
anon, "analytic" in the context of this thread does not mean that it's a smooth function which can be developed into a power series :/

Anonymous

Anonymous Sat 12 Feb 2022 11:09:04 No.14220452 Report

Quoted By: >>14220601

>>14219256
>I don't know why people even mention numerical methods. They're not interesting. We get it, they work for applicatio
Except for application? Gee i wonder why anyone would have an interest in applications

Anonymous

Anonymous Sat 12 Feb 2022 11:33:21 No.14220487 Report

Quoted By:

>>14218028
Numerical methods are brutal, they have the sophistication of a monkey driving a fire truck.

Anonymous

Anonymous Sat 12 Feb 2022 13:01:54 No.14220601 Report

Quoted By:

>>14220452
>Gee i wonder why anyone would have an interest in applications
Engineers, but this is a science board.

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