I'm upset that some ODEs and PDEs can only have numerical solutions

[K.I.L.E.R]

I don't understand why.
I thought algebraic solutions were the best(taking away accuracy excuse from numerical solutions)?

Can the physicists please tell me if I should spend so much time solving all these ODEs by hand when at the appropriate accuracy I can find the same thing via numerical computing?

 

[satein]

I am not a physicist. But I can say that sometime ODE and PDE cannot yields a closed-form solution on every kind of problems.

In order to achieve a closed-form algebraic solution, sometime, a lot of linearizations must be made which in-turns translucent the problem. Sometime, the ODE and PDE governing equation is in the form of unsolveable (infinity dimension as a result of for example delay). Thus, the approximation method is sometime more appropriate . I am now working on instability of a dynamic system caused by delay which the system governing equation yields what we call DDE (Different Differential Equation), which I found that an algebraic solutions is difficult to be expressed and always ends up cumblesome.

Now, I use a new technique called a semi-discretization method for solving a problem which is an interative method mixing between algebraic and numerical solving.

 

[K.I.L.E.R]

Thanks.
So it really depends on the problem at hand and problems come from the limits that the model has?

 

[satein]

It seems both depend on what kind of problem you are dealing with

One limitation is that some kind of signal or respond cannot be represented using a closed form equation, i.e. step function, discontinuity, or some kind of piecewise linear. Thus, even you are trying to use algebraic to solve you still need to compromise by trade off some accuracy by using an approximation function such as Fouier series (transform), polynomian or Chebyshev. Also, to make PDE and ODE to become such a linear problem in order to be solved by linear algebra method, you still need to make assumptions by neglecting or ignoring some parameters by reducing it to be a constant value or sometime disable it to be zero value (mostly appears in fluid dynamic and heat problem). Thus, only simple problem is very efficient for algebraic solver but I am agree with you the algebraic solver is high accuracy and, yes, speedwise but painful at the same time.

However, on a big scale dynamic model, algebraic method is sometime not appropriate as we seem not be able to produce such a function to accurately describ the model, thus reduced form model with discrete degree of freedom is introduced to aid our pain (yes, approximation comes to play again). It's a long long long story.

On engineering problems, mostly we feel alright to trade off an accuracy within +/- 5% (depend on what kind of problem) if the numerical method can give us an answer of what we want to know. It is better than sitting in pain without noting to discuss

Hope this help, and hope you are lucky on solving your problem. Keep the keyword in mine that it is better get a good approximated solution, but talkable, than a complex algebraic one.

Edit: correcting some typo

 

[K.I.L.E.R]

Thank you.
You've made me feel better about using numerical solutions to complex problems.

Usually I ignore numeric solutions to mathematical problems because numeric solutions never give you exactly what you want. Thankfully I'm wrong, otherwise I'd be spending the next 50 years solving sets of complex equations.

 

[satein]

There is no one right no one wrong here :smile: It is just us who have more to learn. The only way is to keep learning. I still insist you to keep on seeking the way to algebraically solve your problem but at the moment just trade it with a numerical one. It is still a long way for me too to learn on what I really have to do to complete my work. Someone said research is what we don't know what we do until it is done