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Lec 20 MIT 18.086 Mathematical Methods for Engineers II

The following content is provided by MIT OpenCourseWare under a Creative Commons license. Additional information about our license and MIT OpenCourseWare in general, is available at ocw.mit.edu. PROFESSOR: I thought I would, in this last lecture before the break, speak about one specific topic. It's often referred to as a fast Poisson solver, so what does Poisson mean? Poisson means Laplace's equation. So, this is the five-point Laplacian, which could be some other discrete Laplace matrix, but let's take the one we know. So we're in two dimensions, and you use Poisson's name, instead of Laplace's name, when there's a non-zero right-hand side. So otherwise, it's a Laplace solver, but here Poisson. OK. So, remember that right-hand side comes from maybe a right-hand side f of x, y in the differential equation, but it also comes from non-zero boundary conditions, because non-zero boundary conditions, when the five points hit a boundary, that kn...
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Lec 20 MIT 18.085 Computational Science and Engineering I

should we have a go on lecture 20 I hate to interrupt all these great conversations uh I promise to stop after an hour uh um so this is lecture 20 which is where the main theme is the finite element method but I have uh continuation completion to to to do of lecture 19 and the you'll see the link right away uh when finite elements come I just indicated up here section 36 is where this calculus of variations is summarized and then finite elements you'll easily identify that section further along uh but this was where we were so at the end of lecture 19 we're looking for the best U of x to make this quantity small and we identified the key principle is move away a little from from U of X move away by some probably I I'm thinking of v ofx as a small perturbation so V ofx I'm thinking of as a small function in any direction positive negative and our quantity would become this but this is supposed if if U is the winner any V is supposed to give us something...
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Lec 20 MIT 18.085 Computational Science and Engineering I, Fall 2008

The following content is provided under a Creative Commons License. Your support will help MIT OpenCourseWare continue to offer high quality education resources for free. To make a donation, or to view additional material from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR STRANG: Okay. Hi. So, our goal, certainly, to reach next week is partial differential equations. Laplace's equation. Additional topics still to see clearly in 1-D. So one of those topics, these will both come today, one of those topics is the idea, still in the finite element world, of element matrices. So you remember, we saw those, that each bar in the truss could give a piece of A transpose A that could be stamped in, or assembled, to use the right word -- I think assembled is maybe used more than stamped in, but both okay -- into K. For graphs, an edge in the graph gave us a little [1, -1; -1, 1] matrix that could be stamped in. And now we want to see, how does that ...
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Lec 20 MIT 18.03 Differential Equations, Spring 2006

Okay, those are the formulas. You will get all of those on the test, plus a couple more that I will give you today. Those will be the basic formulas of the Laplace transform. If I think you need anything else, I'll give you other stuff, too. So, I'm going to leave those on the board all period. The basic test for today is to see how Laplace transforms are used to solve linear differential equations with constant coefficients. Now, to do that, we're going to have to take the Laplace transform of a derivative. And, in order to make sense of that procedure, we're going to have to ask, I apologize in advance, but a slightly theoretical question, namely, we have to have some guarantee in advance that the Laplace transform is going to exist. Now, how could the Laplace transform fail to exist? Can't I always calculate this? And the answer is, no, you can't always calculate it because this is an improper integral. I'm integrating all the way up to infi...
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