PARTIAL DIFFERENTIAL EQUATIONS

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PARTIAL DIFFERENTIAL EQUATIONS
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Introduction

• Given a function u that depends on both x and y,
  the partial derivatives of u w.r.t. x and y are

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• 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.
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• 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

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• 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.

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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)
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Elliptic Equations

• Typically used to characterize steady-state
  distribution of an unknown in two spatial
  dimensions.

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

(Laplace equation)
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Parabolic Equations

• Heat conduction

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

• Finite Difference

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

• First Derivative
• Second Derivative
• Third Derivative

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Solution

• Finite Element

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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.

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