I need to learn how to solve Laplace Equations by hand by my exam day....HELP!

[Bulldozer]

Have you tried searching the web for info on PDEs?

I dropped DiffEQ after about three weeks the first time I enrolled in it as I was simply not getting it.

I took it again with a different prof and actually enjoyed the course. I didn't think it was too tough at all. We covered all of the topics mentioned in this thread. The prof makes all the difference in the world taking a course like this.

There is nothing worse than a textbook with too few examples. I also can't stand it when the book has only one example and the prof chooses to go through that in class instead of creating an example of his/her own.

Most likely a few searches on google will find the info you need.

 

[beer]

Originally posted by: Triumph

Originally posted by: Elemental007

Originally posted by: Vespasian
I've never heard of a introductory differential equations course that covers ODEs, PDEs, and boundary value problems. That's insane.

We did first, second, and higher order linear homogenous and inhomogenous linear equations.
Then we did power series expansions about an ordinary and singular point, including all possible conditions.
We then went into detail for the heat and wave equation for one dimension. And then we started solving fourier series and then laplace equations.

It's insane.

EDIT: Never allowed to touch a calculator or a computer.

wtf? sounds like your teacher doesn't know the standard american curriculum. we didn't do any non-homogeneous equations. series was covered under multivariable calculus, not diffeq.

Laplace transforms are awesome, but there's a reason that there are solution tables for them readily available. As an integral excercise, it really depends on how hard the equation is to begin with, I guess...

Well
my prof got all his degrees, including his ph.D, from the University of Moscow I never assumed for a second he would follow any 'american' cirriculum....

 

[RaynorWolfcastle]

how much do you know about complex variable calculus, that could help a lot. Do the words Jordan's lemma and Cauchy-Goursat theorem mean anything to you?

 

[michaelh20]

My TA is a first year Ph.D candidate from China. I can barely understand him.

Lol.. good luck.. I had to endure this kind of stupidity when I was in school before. They should just plain *fire* people like this because they are truly hopeless as teachers. Can't understand a word? They aren't worth didly squat. I always love the "What did he say?" expressions from all the other people. Plus then they throw hugely complicated things at the class, such that everyone sinks in the mud even further. The most you can hope for is that the prof. you can't understand at all is at least friendly and smiles as you walk out the door, having learned nothing at all.

What a crock. I always tried to walk out on teachers like this and reschedule the class if possible in a semester with a real english speaker.

The only reason that people like this manage to get in the door is to do research for someone else, or for the school, regardless of how crappy a teacher they are.

 

[RaynorWolfcastle]

here are some last minute pointers
remember that
F(s) = int( f(t)*exp(-st) dt) from 0 to ??

if f(t) has no singularities this is an easy integral

ie. if f(t) is a constant K then we have
F(s) = lim (a-> ??) K*[exp(-s*0)/s + K*exp(-s*a)] = K/s

if (t) is a polynomial, you can do it term by term by doing integration by parts. I won't bore you with the details but for a polynomial f(t) = t^n you get F(s) = n!/s^(n+1)

sin's and cos's (as well as their hyperbolic counterparts) also shouldn't be a problem, again do it by parts. For example, for f(t) = sin(bt)
F(s) = b/(s^2+b^2). If you know a bit of complex calculus, you can solve the integral int(exp(ibt+st)) and easily obtain the transform for cos(bt) and sin(bt): Re(int) and Im(int) respectively.

You might have to understand some transforms for generalized functions such as the Dirac delta or the step function. Just memorize those, they're easy to remember anyways.

Convolution is a major pain in the ass, don't use it if at all possible. Otherwise, memorize the form and try applying it.

Also, the nth derivative of f(t) becomes s^nF(s) - s^(n-1)f(0) - s^(n-2)f'(n) - ... - d^(n-1)f/(dt)^(n-1) evaluated at 0 in the S-domain.

For anything that has a singularity on the real axis (ie f(t) = 1/t) you need contour integrals in the complex plane so I'm guessing he hasn't shown you that.

The Laplace inverse is, at your level, just going to be identifying the forms that I stated above and reversing the operation. You'll likely be using a lot of partial fractions, so make sure you revise those.


The PDE's you've learned in your course are probably nothing too complicated here are the 2 cases you'll robably see:
1) take the Laplace with respect to one variable to obtain an ODE in the S-domain with respect to the other variable. Solve using the techniques you know, take the Laplace inverse to get the answer in the time domain. Plug in ICs and BVPs to get rid of constants and you're done. This is how you'll solve the heat equation for simple cases like a heated bar.

2) Assume the variables can be separated, that is, that you can obtain f(x,t) = F(x)G(t). Once you make that assumption there are a lot of things that simplify out pretty nicely and you should be able to obtain a solution using a Fourier expansion for the different modes of vibration.


Fourier expansions are 1/(2Pi)*int(f(t)*exp(-iwt) dt) from -?? to ?? which is easy to evaluate in the complex plane. More than likely however, you can just find the expansion in the region you're looking at since you'll have a string of finite length and of fixed ends and you won't have any singularites.

If you still have time to look at another book, check out Nagle, Saff, and Snider it covers all these topics in detail.

Anyhow, I hope this helps you, let me know if you want a little more detail about something and best of luck on your final


BTW, we did almost everything you covered in our introduction to DE Eq. class though we didn't solve too many heat equations. Although, the dude teaching us did go nuts with the series stuff though and shoed us a ton of stuff with gamma and bessel functions.