Non-linear partial differential equations

[PlaneCrazy]

I'm hoping that someone could refer me to text books in advance calculus or PDEs. I have at least three brilliant maths text book, but all of them stop at elliptical, hyperbolical, Laplace and "conduction"-type PDE's. I am in need of finding methods to solve a non-linear PDE, preferably by transforming into a linear PDE (assuming I can transform the solution back into the original variables and/or time/space reference), or other strategies that may suit my particular need.

I know that this is the kind of stuff that Ph.D's are made of, but right now this is more of a hobby for me. The result could, however, forward my career in mechanical engineering as well, as soon as I find out how...

 

[PlaneCrazy]

BTW, if this thread or forum is dedicated to "electronics only" maths, please ignore this.

 

[MrAl]

Hello there,

The jurisprudence of electronic laws include PDE's once in a while (har har) as in the 'Telegraphers Equation".
If you can get away with numerical methods this might help:
https://ocw.mit.edu/courses/aeronau...equations-sma-5212-spring-2003/lecture-notes/

 

[PlaneCrazy]

Great link!!!

It has all the Laplace, elliptical, and others I've mentioned...

I'm trying to not use full spatial discretisation. BTW, some aerodynamic theory was based on electromagnetism, so as far as I'm concerned, math is math, right?

 

[MrAl]

Hi again,

Well after all this is a physics and math sub section, and we often talk about math subjects without concern for a specific electrical problem, knowing that math plays a key role in all the physical laws that certainly include electron current flow or field theory. Most of us here however havent done this kind of thing in many years...last time i used finite element analysis for example was back when the dinosaurs were still roaming the earth (he he).

What i do like about numerical analysis though is that it often seems less abstract than pure mathematics, so when i get very nearly the same solution using pure theory and a numerical method too i tend to feel more confident about the accuracy of the result. It's too easy to make a mistake so it's always good to have two ways to do something so that the results can be verified, even if the second way takes much longer to calculate. If i calculate 0.12345 with one method for a given set of conditions and it takes me one second to do that, then calculate 0.23456 with the second method, i made a mistake somewhere, but if i get 0.12346 with the second method even if it takes 10 minutes to calculate with the second slower numerical method, i can be much more certain of the original result even though the numerical result isnt as accurate (usually the way it turns out). Not a guarantee but still helps.

 

[PlaneCrazy]

Except that numerical method make us lazy. We often forget just how inefficient these methods are because of the exponential development rate of the PC, but would you believe that one of my lecturers once lamented that we can save ourselves so much time by developing analytical expressions as far as possible? He wanted to demonstrate a program he wrote to determine the roots of an implicit expression or something. To roots are where somewhre between one point being positve and the other being negative - that condition took him two lines of coding after several tries because he forgot that the product of two values is negative and only negative if one value is positive and the other negative...

Numerical methods are great for solving all kinds of stuff, but the solution methods are inefficient. Using better equations, one can reduce a numerical code's solution time from literally several weeks to a few minutes. "Fluent" is excepted as industry-standard fluid mechanics solver, but it is at best a validation tool and utterly useless as a design tool, because it takes so long and makes a whole bunch of simplifying assumptions. "Patran" is accepted as the industry-standard structural FEM package, despite being inferior in far too many respects to "Dytran" both of which are still pretty useless for a design tool.

Besides, I find mathematics universal and beautiful, and will further my stiudies as soon as I get the opportunity. But I would like to know that I won't be wasting my time.

Watch this space for some breath-taking expressions from yours truly.

 

[MrAl]

Hello again,

Oh dont get me wrong, i would prefer an analytical method over numerical just like the next guy, but many times the analytical solution is either too hard to find or may not exist. I wasnt really talking about this though, i was just mentioning that numerical methods have their place. But now that you bring it up, there are numerical methods for non linear electrical networks that may never be solved at all without them. I quickly realized how futile some of these analytical methods could be after studying some non linear control systems with the intent on coming up with an analytic method to determine stability. One system out of ten works out. Buy anyway i would still like to see what you have to offer here and await your 'breath taking expressions'

 

[PlaneCrazy]

Actually, I forgot to mention that, as far as structural mechanics go, I don't mind full discretisation of each/all member(s), since their geometry is fully defined and everything else is (relatively) non-load bearing. But when it comes to fluid mechanics, I prefer boundary element discretisation over full spatial discretisation simply because they take less time and, should the structure deform due to the fluid dynamic loads, the changes to the boundary elements would be far fewer than the changes to discretised space, hence taking less time...

So, here is an example of an equation which I would love to solve using the boundary element method (BEM). Equations like this one serve as proof that we engineers are mathematically inapt, since we rely so heavily on (not necesserally the best) tools. This is why I'm looking for advanced mathematical tex books, or to study mathematics before pursuing further post-grad engineering.

Read it and weep.

 

[PlaneCrazy]

Believe me, derivation of the above is no easy feat, unless you bypass pages and pages and pages of writing with that popular maths-statement "it can easily be shown that..."

 

[MrAl]

Hi again,

Well it certainly looks like you have your work cut out for you
I wish you the best of luck, and if you find any useful equations for general electronics applications or even robotics or control theory i wouldnt mind taking a look.

 

[fernandoagv]

well imagine considering the first equation curve as a flat line so a secondary differential equation fits in your requirements

 

[fernandoagv]

maybe considering a first equation curve as a flat line base so a secondary equation fits the requirements...
I tought digitalization got rid of this...