SE301:Numerical Methods Topic 9 Partial Differential Equations

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SE301Numerical MethodsTopic 9Partial
Differential Equations

• Dr. Samir Al-Amer
• Term 071

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Lect 27 Partial Differential Equations

• Partial Differential Equations (PDE)
• What is a PDE
• Examples of Important PDE
• Classification of PDE

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Partial Differential Equations
813
A partial differential equation (PDE) is an
equation that involves an unknown function and
its partial derivatives.
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Notation
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Linear PDEClassification
813
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Representing the solution of PDE(two independent
variables)

• Three main ways to represent the solution

T5.2
t1
T3.5
x1
Different curves are used for different values of
one of the independent variable
Three dimensional plot of the function T(x,t)
The axis represent the independent variables. The
value of the function is displayed at grid points
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Heat Equation
Different curve is used for each value of t
ice
ice
Temperature at different x at t0
Temperature
x
Thin metal rod insulated everywhere except at
the edges. At t 0 the rod is placed in ice
Position x
Temperature at different x at th
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Heat Equation
Temperature T(x,t)
Time t
ice
ice
x
t1
Position x
x1
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Linear Second Order PDEClassification
814
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Linear Second Order PDEExamples ( Classification)
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Classification of PDE

• Linear Second order PDE are important set of
  equations that are used to model many systems in
  many different fields of science and engineering.
  
• Classification is important because
• Each category relates to specific engineering
  problems
• Different approaches are used to solve these
  categories

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Examples of PDE

• PDE are used to model many systems in many
  different fields of science and engineering.
• Important Examples
• Wave Equation
• Heat Equation
• Laplace Equation
• Biharmonic Equation

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Heat Equation
The function u(x,y,z,t) is used to represent the
temperature at time t in a physical body at a
point with coordinates (x,y,z) .
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Simpler Heat Equation
x
u(x,t) is used to represent the temperature at
time t at the point x of the thin rod.
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Wave Equation
The function u(x,y,z,t) is used to represent the
displacement at time t of a particle whose
position at rest is (x,y,z) . Used to model
movement of 3D elastic body
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Laplace Equation
Used to describe the steady state distribution of
heat in a body. Also used to describe the steady
state distribution of electrical charge in a body.
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Biharmonic Equation
Used in the study of elastic stress.
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Boundary conditions for PDE

• To uniquely specify a solution to the PDE, a set
  of boundary conditions are needed.
• Both regular and irregular boundaries are possible

t
region of interest
x
1
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The solution Methods for PDE

• Analytic solutions are possible for simple and
  special (idealized) cases only.
• To make use of the nature of the equations,
  different methods are used to solve different
  classes of PDE.
• The methods discussed here are based on finite
  difference technique

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Elliptic Equations

• Elliptic Equations
• Laplace Equation
• Solution

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Elliptic Equations
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Laplace Equation

• Laplace equation appears is several
  engineering problems such as
• Studying the steady state distribution of heat in
  a body
• Studying the steady state distribution of
  electrical in a body

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Laplace Equation

• Temperature is function of the position (x and y)
• When no heat source is available ?f(x,y)0

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Solution Technique

• A grid is used to divide region of interest
• Since the PDE is satisfied at each point in the
  area, it must be satisfied at each point of the
  grid.
• A finite difference approximation is obtained at
  each grid point.

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Solution Technique
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Solution Technique
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Solution Technique
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Example

• It is required to determine the steady state
  temperature at all points of a heated sheet of
  metal. The edges of the sheet are kept at
  constant temperature 100,50, 0 and 75 degrees.

100
50
75
The sheet is divided by 5X5 grids
0
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Example
Known To be determined
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First equation
Known To be determined
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Example
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Another Equation
Known To be determined
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Solution The rest of the equations
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Convergence and stability of solution

• Convergence
• The solutions converge means that the solution
  obtained using finite difference method
  approaches the true solution as the steps
  approaches zero.
• Stability
• An algorithm is stable if the errors at each
  stage of the computation are not magnified as the
  computation progresses.

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Parabolic Equations

• Parabolic Equations
• Heat Conduction Equation
• Explicit Method
• Implicit Method
• Cranks Nicolson Method

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Parabolic Equations
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Parabolic Problems
ice
ice
x
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First order Partial derivative Finite Difference
Forward difference Method
Backward difference Method
Central difference Method
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Finite Difference Methods
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Finite Difference MethodsNew Notation
Superscript for t-axis And Subscript for
x-axis Til-1T(x,t-?t)
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Solution of the PDE
t
x
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Solution of the Heat Equation
Two solutions to the Parabolic Equation (Heat
Equation) will be presented 1. Explicit Method
Simple, Stability Problems 2.
Crank-Nicolson Method involves solution of
Tridiagonal system of equations, stable.
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Explicit Method
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Explicit MethodHow do we compute
u(x,tk)
u(x-h,t) u(x,t)
u(xh,t)
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Explicit MethodHow do we compute
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Explicit Method
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Crank-Nicolson Method
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Explicit MethodHow do we compute
u(x-h,t) u(x,t)
u(xh,t)
u(x,t - k)
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Crank-Nicolson Method
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Crank-Nicolson Method
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Outlines

• Examples
• Explicit method to solve Parabolic PDE
• Cranks-Nicholson Method

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Heat Equation
ice
ice
x
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Example 1
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Example 1 (cont.)
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Example 1
0
0
t1.0
0
0
t0.75
t0.5
0
0
t0.25
0
0
0
0
t0
Sin(0.25p)
Sin(0. 5p)
Sin(0.75p)
x0.0
x1.0
x0.25
x0.5
x0.75
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Example 1
0
0
t1.0
0
0
t0.75
t0.5
0
0
t0.25
0
0
0
0
t0
Sin(0.25p)
Sin(0. 5p)
Sin(0.75p)
x0.0
x1.0
x0.25
x0.5
x0.75
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Example 1
0
0
t1.0
0
0
t0.75
t0.5
0
0
t0.25
0
0
0
0
t0
Sin(0.25p)
Sin(0. 5p)
Sin(0.75p)
x0.0
x1.0
x0.25
x0.5
x0.75
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Remarks on Example 1
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Example 1
0
0
t0.10
0
0
t0.075
t0.05
0
0
t0.025
0
0
0
0
t0
Sin(0.25p)
Sin(0. 5p)
Sin(0.75p)
x0.0
x1.0
x0.25
x0.5
x0.75
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Example 1
0
0
t0.10
0
0
t0.075
t0.05
0
0
t0.025
0
0
0
0
t0
Sin(0.25p)
Sin(0. 5p)
Sin(0.75p)
x0.0
x1.0
x0.25
x0.5
x0.75
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Example 1
0
0
t0.10
0
0
t0.075
t0.05
0
0
t0.025
0
0
0
0
t0
Sin(0.25p)
Sin(0. 5p)
Sin(0.75p)
x0.0
x1.0
x0.25
x0.5
x0.75
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Example 2
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Example 2 Crank-Nicolson Method
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Example 2Crank-Nicolson Method
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Example 2
0
0
t1.0
0
0
t0.75
t0.5
0
0
u1 u2 u3
t0.25
0
0
0
0
t0
Sin(0.25p)
Sin(0. 5p)
Sin(0.75p)
x0.0
x1.0
x0.25
x0.5
x0.75
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Example 2
0
0
t1.0
0
0
t0.75
t0.5
0
0
u1 u2 u3
t0.25
0
0
0
0
t0
Sin(0.25p)
Sin(0. 5p)
Sin(0.75p)
x0.0
x1.0
x0.25
x0.5
x0.75
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Example 2
0
0
t1.0
0
0
t0.75
t0.5
0
0
u1 u2 u3
t0.25
0
0
0
0
t0
Sin(0.25p)
Sin(0. 5p)
Sin(0.75p)
x0.0
x1.0
x0.25
x0.5
x0.75
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Example 2Crank-Nicolson Method
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Example 2Second Row
0
0
t1.0
0
0
t0.75
u1 u2 u3
t0.5
0
0
t0.25
0
0
0.2115 0.2991 0.2115
0
0
t0
Sin(0.25p)
Sin(0. 5p)
Sin(0.75p)
x0.0
x1.0
x0.25
x0.5
x0.75
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Example 2
The process is continued until the values of
u(x,t) on the desired grid are computed.
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Remarks

• The explicit metod
• one need to select small k to ensure stability
• Computation per point is very simple but many
  points are needed.
• Cranks Nicolson
• Requires solution of Tridiagonal system
• Stable (larger k can be used).