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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.

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partial differential equations
Partial Differential Equations
  • Introduction
    • Adam Zornes, Deng Li
  • Discretization Methods
    • Chunfang Chen, Danny Thorne, Adam Zornes
what do you stand for
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
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
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
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
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
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
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
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
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
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
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
coupled eigenvalue problem
Coupled Eigenvalue Problem
  • Structure/Acoustic Coupled system
3d coupled problem
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 un coupled problem
2D Un-coupled Problem
  • Acoustic Problem
2d un coupled problem19
2D Un-coupled Problem
  • Structure Problem

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