Download

Partial Differential Equations






Advertisement
/ 20 []
Download Presentation
Comments
selena
From:
|  
(1024) |   (0) |   (0)
Views: 137 | Added:
Rate Presentation: 1 0
Description:
Partial Differential Equations. Introduction Adam Zornes, Deng Li Discretization Methods Chunfang Chen, Danny Thorne, Adam Zornes. What do You Stand For?. A PDE is a P artial D ifferential E quation This is an equation with derivatives of at least two variables in it.
Partial Differential Equations

An Image/Link below is provided (as is) to

Download Policy: Content on the Website is provided to you AS IS for your information and personal use only and may not be sold or licensed nor shared on other sites. SlideServe reserves the right to change this policy at anytime. While downloading, If for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.











- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -




Partial differential equations l.jpgSlide 1

Partial Differential Equations

  • Introduction

    • Adam Zornes, Deng Li

  • Discretization Methods

    • Chunfang Chen, Danny Thorne, Adam Zornes

What do you stand for l.jpgSlide 2

What do You Stand For?

  • A PDE is a Partial Differential Equation

  • This is an equation with derivatives of at least two variables in it.

  • In general, partial differential equations are much more difficult to solve analytically than are ordinary differential equations

What does a pde look like l.jpgSlide 3

What Does a PDE Look Like

  • Let u be a function of x and y. There are several ways to write a PDE, e.g.,

    • ux + uy = 0

    • du/dx + du/dy = 0

The baskin robin s esq characterization of pde s l.jpgSlide 4

The Baskin Robin’s esq Characterization of PDE’s

  • The order is determined by the maximum number of derivatives of any term.

  • Linear/Nonlinear

    • A nonlinear PDE has the solution times a partial derivative or a partial derivative raised to some power in it

  • Elliptic/Parabolic/Hyperbolic

Six one way l.jpgSlide 5

Six One Way

  • Say we have the following: Auxx + 2Buxy + Cuyy + Dux + Euy + F = 0.

  • Look at B2 - AC

    • < 0 elliptic

    • = 0 parabolic

    • > 0 hyperbolic

Or half a dozen another l.jpgSlide 6

Or Half a Dozen Another

  • A general linear PDE of order 2:

  • Assume symmetry in coefficients so that A = [aij] is symmetric. Eig(A) are real. Let P and Z denote the number of positive and zero eigenvalues of A.

    • Elliptic: Z = 0 and P = n or Z = 0 and P = 0..

    • Parabolic: Z > 0 (det(A) = 0).

    • Hyperbolic: Z=0 and P = 1 or Z = 0 and P = n-1.

    • Ultra hyperbolic: Z = 0 and 1 < P < n-1.

Elliptic not just for exercise anymore l.jpgSlide 7

Elliptic, Not Just For Exercise Anymore

  • Elliptic partial differential equations have applications in almost all areas of mathematics, from harmonic analysis to geometry to Lie theory, as well as numerous applications in physics.

  • The basic example of an elliptic partial differential equation is Laplace’s Equation

    • uxx - uyy = 0

The others l.jpgSlide 8

The Others

  • The heat equation is the basic Hyperbolic

    • ut - uxx - uyy = 0

  • The wave equations are the basic Parabolic

    • ut - ux - uy = 0

    • utt - uxx - uyy = 0

  • Theoretically, all problems can be mapped to one of these

What happens where you can t tell what will happen l.jpgSlide 9

What Happens Where You Can’t Tell What Will Happen

  • Types of boundary conditions

    • Dirichlet: specify the value of the function on a surface

    • Neumann: specify the normal derivative of the function on a surface

    • Robin: a linear combination of both

  • Initial Conditions

Is it worth the effort l.jpgSlide 10

Is It Worth the Effort?

  • Basically, is it well-posed?

    • A solution to the problem exists.

    • The solution is unique.

    • The solution depends continuously on the problem data.

  • In practice, this usually involves correctly specifying the boundary conditions

Why should you stay awake for the remainder of the talk l.jpgSlide 11

Why Should You Stay Awake for the Remainder of the Talk?

  • Enormous application to computational science, reaching into almost every nook and cranny of the field including, but not limited to: physics, chemistry, etc.

Example l.jpgSlide 12

Example

  • Laplace’s equation involves a steady state in systems of electric or magnetic fields in a vacuum or the steady flow of incompressible non-viscous fluids

  • Poisson’s equation is a variation of Laplace when an outside force is applied to the system

Poisson equation in 2d l.jpgSlide 13

Poisson Equation in 2D

Example cfd l.jpgSlide 14

Example: CFD

Coupled eigenvalue problem l.jpgSlide 15

Coupled Eigenvalue Problem

  • Structure/Acoustic Coupled system

3d coupled problem l.jpgSlide 16

3D Coupled Problem

Ω0 : a three-dimensional acoustic region,

S0 : a plate region,

Γ0=∂Ω0\S0 : a part of the boundary of the acoustic field,

∂S0 : the boundary of the plate,

P0 : the acoustic pressure in Ω0,

U0 : the vertical plate displacement,

c : the sound velocity,

ρ0 : the air mass density,

D : the flexural rigidity of plate,

ρ1 : the plate mass density,

n : the outward normal vector on ∂Ω from Ω0, and

σ : the outward normal vector on ∂S0 from S0.

2d coupled problem l.jpgSlide 17

2D Coupled Problem

2d un coupled problem l.jpgSlide 18

2D Un-coupled Problem

  • Acoustic Problem

2d un coupled problem19 l.jpgSlide 19

2D Un-coupled Problem

  • Structure Problem

,

.

Homework make the procese how to get the eigenvalue of structure problem on previous page l.jpgSlide 20

HomeworkMake the procese how to get theeigenvalue of structure problem(on previous page).


Copyright © 2014 SlideServe. All rights reserved | Powered By DigitalOfficePro