Why do we derive analytic solutions

[ice109]

given some complex model that does not generally have analytic solutions why do we search for, and solve for, cases where there are analytic solutions? considering at this point much simulation can be done using numerical methods on computers what is the point?

take for example a pendulum which is a nonlinear oscillator. we can only solve this problem analytically in the limit that the pendulum sweeps out some small angle theta. why in the world is that even taught considering an iterative difference method can solve the nonlinear problem more accurately than the simplified model models reality?

if I'm wrong about this please correct me, I've obviously assumed the numerical soln to the nonlinear problem can be made arbitrarily accurate.

 

[swraman]

I think its just thought so that you will understand what you are solving for in future jobs you have, when you use a computer to solve them. Because you do learn a lot more when you solve thinks by hand, a whole lot more.

 

[Andy Resnick]

This is a very important question to ask. It's intimately related to the question "Why use mathematical models?", and understanding the answer requires answering "Why have model systems?"

Ice109, let me play Devil's advocate and pose two questions to you:

1) What is the point of having a mathematical model?
2) How do you know your model is correct?

Let's use your example- "a pendulum which is a nonlinear oscillator". Already, you assume a mathematical model *with a specific form* which has abstracted the dynamical behavior of a real object. This represents a gigantic conceptual leap which is not often appreciated. Because, as you must admit, your model system will not precisely predict the dynamical behavior of my object for all time. So what good is your model, if you know already it is invalid?

 

[ice109]

i don't think you, Andy, understood my question. I'm not saying we shouldn't have models, I'm saying deriving closed forms solutions is a waste of time.

 

[Cyrus]

i don't think you, Andy, understood my question. I'm not saying we shouldn't have models, I'm saying deriving closed forms solutions is a waste of time.

And so if there is a solution in a book and your boss says to find the answer. Your going to explain to him how you wasted your time writing a code when you could have just looked it up?

We do it so we can look at trends to streamline the simulation process. Try doing a design of experiment (DOE) analysis using a simulation that runs about an hour min each for many design iterrations. We had to do that for our senior design project. Its a HUGE waste of time running simulations all day long.

Don't be rude to Andy when he's trying to help you.

 

[Integral]

Given a closed from solution you can do extensive further analysis. For example, You can look at the "solution" at ANY point independently of any other point. You can determine a closed form expression for the rate of change, you can locate interesting points like roots, maximum, minimum in general. A closed form solution is just more useful.

A numerical solution requires that you iterate from boundary or initial conditions to a point in the interior. This is simply not as informative or as useful as a closed form solution. While you can get some interesting pictures with a numerical solution there is much more that you cannot do.

 

[D H]

given some complex model that does not generally have analytic solutions why do we search for, and solve for, cases where there are analytic solutions? considering at this point much simulation can be done using numerical methods on computers what is the point?

Andy already gave you one very good answer here:

2) How do you know your model is correct?

Analytic solutions are one way of demonstrating correctness and proving incorrectness. Take your example of a nonlinear pendulum. Suppose you are asked to develop a tool to model the evolution of such a system given a set of initial conditions. Rather than developing an analytic solution, you use a numerical techniques to propagate the system forward in time. How do you know that

1. You have the correct equations of motion? In the real world, this is called validation. In validating a system one must ask questions such as "Is this the right model?"
2. You have implemented the equations of motion correctly? In the real world, this is called verification. In verifying a system one must ask questions such as "How do I know I did this correctly?"

Analytic solutions help (or hurt) in verification. In this case you can run your nonlinear pendulum model multiple times against initial conditions with ever decreasing energy. The resultant outputs from this series of runs should move toward the analytical small angle approximation solution as the energy decreases. There is something wrong with your simulation if your simulation does not do that.

Analytic solutions also help (or hurt) in validation. Suppose in some review of your model you tell the reviewers that you had to use numerical simulation because "we can only solve this problem analytically in the limit that the pendulum sweeps out some small angle theta." Suppose one of the reviewers is well-versed in mathematics and says "That's not true! There is an analytic solution to the nonlinear pendulum. It is expressed in terms of elliptical integrals of the first kind. You should have used the analytic solution here." How do you respond?

 

[Andy Resnick]

i don't think you, Andy, understood my question. I'm not saying we shouldn't have models, I'm saying deriving closed forms solutions is a waste of time.

So... are you just complaining about your homework?

 

[ice109]

So... are you just complaining about your homework?

yes.

 

[vanesch]

As others pointed out, analytical solutions have the following advantages:
- they can sometimes give you more insight (that's only true if the analytic solution is relatively simple: if you have analytical expressions on 20 pages generated by Mathematica for instance, I don't know how much more insight you get that way)
- they are usually a solution to a *whole set* of solutions at once (symbolic parameters), and not a single point in the solution space: so you learn stuff about the whole solution space, and not just about a single point in it
- they are a very very good check on numerical solutions

But there's an important point that has not been addressed here: you NEED analytical solutions to set up numerical solutions! Finite-element systems (and variants) are nothing else but a lot of analytical solutions in small volumes, and the "numerical" part is nothing else but fitting together the boundary conditions of all these analytical solutions.

 

[dx]

Why did you start the same thread again in a different place?

 

[D H]

Why did you start the same thread again in a different place?

I suspect two reasons: (1) He didn't get the answer he wanted the first time around. (2) He really hates doing math homework and wants sympathy.

 

[maverick_starstrider]

The exact same things always bugged me which is why I went into Computational Physics (where everything is numerical). Of course you pretty much always need to derive a simplified mean theory solution or the likes for comparision so you end up having to find an analytic solution anyways.

 

[Meir Achuz]

given some complex model that does not generally have analytic solutions why do we search for, and solve for, cases where there are analytic solutions? considering at this point much simulation can be done using numerical methods on computers what is the point?

take for example a pendulum which is a nonlinear oscillator. we can only solve this problem analytically in the limit that the pendulum sweeps out some small angle theta. why in the world is that even taught considering an iterative difference method can solve the nonlinear problem more accurately than the simplified model models reality?

if I'm wrong about this please correct me, I've obviously assumed the numerical soln to the nonlinear problem can be made arbitrarily accurate.

I have not read the other responses, but will give my reply to your question.
Many so called "analytic" solutions are not really analytic. For instance,
finding a solution in terms of Bessel functions is really a numerical solution that is common enough to be named after a famous mathematician. Before computers, "named numerical solutions" were thought of as "analytical solutions" because hundreds of arithmeticians had laboriously tabulated them. Now it is often easier to just compute the numerical work without searching for named functions, but it takes a while for this to be appreciated.
Some named functions, like Bessel, sin, cos, are still useful because they have simple visualizations which can be helpful.

 

[vanesch]

I have not read the other responses, but will give my reply to your question.
Many so called "analytic" solutions are not really analytic. For instance,
finding a solution in terms of Bessel functions is really a numerical solution that is common enough to be named after a famous mathematician. Before computers, "named numerical solutions" were thought of as "analytical solutions" because hundreds of arithmeticians had laboriously tabulated them. Now it is often easier to just compute the numerical work without searching for named functions, but it takes a while for this to be appreciated.
Some named functions, like Bessel, sin, cos, are still useful because they have simple visualizations which can be helpful.

I don't agree. Bessel functions aren't called analytic because you can find their values tabulated in Abramowitz and Stegun. They are called analytic because we know several, eh, analytic properties of them. We know a lot of relationships about them, look at http://mathworld.wolfram.com/BesselFunctionoftheFirstKind.html for instance.

I would agree with you that if the only thing we knew about it was a table of numerical values, then it would just be a name for a table of numerical values. But analytic means more than that.

 

[Borek]

Assuming you know analytical solution x=f(t) you just plug t and you know the position. Numerical approach means you have to simulate everything from t₀ to t, that can take a lot of time. There is also often a possibility that calculation errors accumulate, so the larger the t, the less likely is your numerical answer to be correct. Analytical is just right.