From: israel@math.ubc.ca (Robert Israel) Subject: Re: Soultion Methods for PDE's Date: 3 Mar 2000 18:11:52 GMT Newsgroups: sci.math,alt.math.recreational Summary: [missing] In article <38BC366C.7FAA8564@rpi.edu>, Andy Littlefield writes: > I need to find some possible solutions methods for two second order > coupled partial differential equations. The equations are: > > A*(d^2u/dx^2) + B*(d^2v/dxdy) + C*(d^2u/dy^2) = D*(d^2u/dt^2) > > E*(d^2v/dy^2) + F*(d^2u/dxdy) + G*(d^2v/dx^2) = H*(d^2v/dt^2) > > where A,B,C,D,E,F,G,H are constant coefficients > u is the displacement in the x-direction > v is the displacement in the y-direction and > t is time Differentiate the second expression with respect to x and y, then substitute for d^2v/dxdy from the first, and you get a fourth order equation involving u only. There will be solutions of the form u = F(a x + b y + c t) (where F is arbitrary) if a,b,c satisfy the equation 2 2 2 2 4 2 2 2 2 4 E D c b - E A a b - E C b + F a b B + G D c a - G A a 2 2 4 2 2 2 2 - G C a b - D c + A c a + C c b = 0 Since this depends only on the squares of a,b,c, this is easy to solve. Substitute the resulting u in the first equation and integrate with respect to x and y to get v. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 ============================================================================== From: Jeffery J. Leader Subject: Re: Spectral methods for PDEs Date: Thu, 21 Dec 2000 00:15:28 -0500 Newsgroups: sci.math.num-analysis Summary: [missing] On Wed, 20 Dec 2000 07:54:14 GMT, r08n@my-deja.com wrote: >Hi there, >I'm looking for a decent tutorial/introduction/concise description on >spectral methods for solution of >PDEs, particularly, nonlinear Schroedinger equation. Is it possible to >find any downloadable documentation for that matter? I would be >grateful for any links. Thank you. I was fortunate to be able to take Spectral Methods with David Gottlieb while at Brown. Fascinating stuff. You might check out the web page of my old graduate school colleague Wai Sun Don who is still at Brown University: http://www.cfm.brown.edu/people/wsdon/home.html His page links to the PseudoSpectral Differentiation Software Package web page: http://www.labma.ufrj.br/~bcosta/PseudoPack2000/Main.html If you're willing to actually pay for info., though, you might want to check out Lloyd Trefethen's new book on Spectral Methods in Matlab from SIAM. (They actually sent me two copies for the price of one--evidently the first one has some missing or misprinted figures. I donated the broken copy to the dept. library.) It's a reasonably good introduction and it's not very expensive.