Numerical methods: relevance

[Fiidweee]

This question is about why I'm requested by my teacher to employ numerical methods instead of analytical ones.

The subject I'm writing about is rocket science. The task is to predict the movement of a rocket in 2 dimensions. I understand this is a maths forum, but we'll get there.
I'm able to construct two acceleration equations:

u, g, A, cw, p (rho), dm/dt are all constants while m (mass of the rocket) and v_y and v_x change depending on the time (speed in vertical and horizontal direction respectively). Since I'm interested in the position functions, I must solve second order differential equations.

I wrote in my project that they are difficult to solve analytically because it's a 'linked/codependent equation system' without really knowing what I was talking about. Basically, I need to understand why it's relevant to use numerical methods instead of analytical ones in this problem.

 

[HallsofIvy]

You say it is difficult to use analytical methods- how difficult would numerical methods be?

 

[Fiidweee]

You say it is difficult to use analytical methods- how difficult would numerical methods be?

I don't know if it is difficult to use analytical methods, I guess that's what I'm trying to figure out.

As a high school student, it was rather difficult to manage.
To carry out the task, I've written a program that manages the large amount of calculational loops. Each step depends on the previous one.
1. The acceleration is calculated in each direction with the equations above.
2. The velocity is calculated with the help of Runge-Kutta 4th order.
3. The position is calculated with the help of forward Euler.

Any help is greatly appreciated as I truly need to try to understand. It was quite a route to take, so why not do it analytically. I don't know how you would do it analytically nonetheless.

 

[JeffM]

First, many differential equations are not currently solvable using analytic methods. The only way to "solve" them so far is approximately by numerical methods.

Second, it may be less work to write a program than to try to find an analytic solution. (Remember that there may not be one to find, and even if there is one, who knows how long it might take to find it.)

Does this answer your question?

 

[Fiidweee]

First, many differential equations are not currently solvable using analytic methods. The only way to "solve" them so far is approximately by numerical methods.

Second, it may be less work to write a program than to try to find an analytic solution. (Remember that there may not be one to find, and even if there is one, who knows how long it might take to find it.)

Does this answer your question?

That depends, is it unsolvable in this case? If I am questioned at the examination, I'll need to be able to answer for this very specific case.

I understand that some differential equations are unsolvable, if not most, but I'm not sure if I'm to say that this would be too difficult for me to manage, or that it is indeed impossible. Saying something is impossible is quite 'extreme' so to speak.

 

[topsquark]

The system of equations is pretty non-linear. That's not to say that analytical solutions don't exist but unless you are a masochist I agree that your best bet is a numerical solution.

-Dan

 

[Deleted member 4993]

The system of equations is pretty non-linear. That's not to say that analytical solutions don't exist but unless you are a masochist I agree that your best bet is a numerical solution.

-Dan

I don't know about that! Beauty - they say, is in the eye of the beholder.

 

[Fiidweee]

So as far as my high school maths are able to contribute to my understanding of what you guys are writing, it's not easily discerned if it's analytically solvable. And if it is, it would take a large amount of time to figure out.

How would one explain why this is? What properties/characteristics of the equations make them difficult to solve analytically? I'm afraid I'll need that spelled out before I really catch on.

 

[Deleted member 4993]

So as far as my high school maths are able to contribute to my understanding of what you guys are writing, it's not easily discerned if it's analytically solvable. And if it is, it would take a large amount of time to figure out.

How would one explain why this is? What properties/characteristics of the equations make them difficult to solve analytically? I'm afraid I'll need that spelled out before I really catch on.

If you do a google search using:

methods of solving non-linear differential equations

You will get 14*10⁷ "hits".

Read some of those to satisfy your query .....

 

[JeffM]

So as far as my high school maths are able to contribute to my understanding of what you guys are writing, it's not easily discerned if it's analytically solvable. And if it is, it would take a large amount of time to figure out.

How would one explain why this is? What properties/characteristics of the equations make them difficult to solve analytically? I'm afraid I'll need that spelled out before I really catch on.

There are classes of differential equations that are known to be analytically soluble. In a course on differential equations, you are taught to recognize those classes and how to apply the correct analytic technique. As far as I know, no one has proved that specific classes cannot possibly have an analytic method of solution. It is just that no one has discovered a method that works. There may be one, but you can't use it until it is found. If you cannot match up your equation to one of the types for which an analytic method of solution is already known, you can either try to develop such a method on your own or use numerical methods to find an approximate solution.

In many practical cases, there is little incentive to find an exact theoretical solution because the parameters of the equation are themselves approximations so the answer will be an approximation anyway. In situations of that sort, the emphasis shifts to determining how good an approximation is needed and how easy is it to get such an approximation.