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PARTIAL DIFFERENTIAL EQUATIONS

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PARTIAL DIFFERENTIAL EQUATIONS Introduction Given a function u that depends on both x and y, the partial derivatives of u w.r.t. x and y are: An equation involving ... – PowerPoint PPT presentation

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Title: PARTIAL DIFFERENTIAL EQUATIONS


1
PARTIAL DIFFERENTIAL EQUATIONS
2
Introduction
  • Given a function u that depends on both x and y,
    the partial derivatives of u w.r.t. x and y are

3
  • An equation involving partial derivatives of an
    unknown function of two or more independent
    variables is called Partial Differential Equation
    (PDE). Examples

The order of a PDE is that of the highest-order
partial derivative appearing in the equation.
4
  • A PDE is linear if it is linear in the unknown
    function and all its derivatives, with
    coefficients depending only on the independent
    variables
  • e.g.
  • x ax bx c 0 linear
  • x t2x linear
  • x 1/x nonlinear

5
  • For linear, two independent variables second
    order equations can be expressed as
  • where A, B and C are functions of x and y and D
    is a function of x, y, u/x and u/y.
  • Above equation can be classified into categories
    in the next slide based on values of A, B, and C.

6
B2 4AC Category Example
lt 0 Elliptic Laplace equation (Steady state with two spatial dimension)
0 Parabolic Heat conduction equation (time variable with one spatial dimension)
gt 0 Hyperbolic Wave equation (time variable with one spatial dimension)
7
Elliptic Equations
  • Typically used to characterize steady-state
    distribution of an unknown in two spatial
    dimensions.

8
Laplace Equation
The PDE as an expression of the conservation of
energy
9
  • Need to reformulate the equation in terms of
    temperature. Use Fouriers Law
  • and
  • substituting back results in

(Laplace equation)
10
Parabolic Equations
  • Heat conduction

Hot
Cool
Heat balance (the amount of heat stored in the
element) over a unit time, Dt
11
Input Output Storage
Dividing by volume of the element (DxDyDz) and Dt
Taking the limit yields
12
Substituting Fouriers Law
Gives
13
Solution
  • Finite Difference

A grid used for the finite difference solution of
elliptic PDEs in two independent variables.
14
Numerical Differentiation using Centred-Finite
Divided Difference
  • First Derivative
  • Second Derivative
  • Third Derivative

15
Solution
  • Finite Element

16
Finite Element Analysis
  • Two interpretations
  • Physical Interpretation
  • The continous physical model is divided into
    finite pieces called elements and laws of nature
    are applied on the generic element. The results
    are then recombined to represent the continuum.
  • Mathematical Interpretation
  • The differentional equation representing the
    system is converted into a variational form,
    which is approximated by the linear combination
    of a finite set of trial functions.

17
Group Assignment
Group Task
Group A Problem 1
Group B Problem 2
Group C Problem 3
Group D Problem 4
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