Partial Differential Equations

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Partial Differential Equations

• Chapter 12
• Burden and Faires

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General 2nd Order Form
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Discriminant
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Classification
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Conic Sections
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Classification
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Elliptical PDE
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Elliptical PDE
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FDM Molecule
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ODE Second Order Formula
xi1
xi
xi-1
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ODE Second Order Central Difference Formulas
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ODE Second Order Central Difference Formulas
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PDE Second Order Central Difference Formulas
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PDE Second Order Central Difference Formulas
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Example Problem
Consider a square plate 0.5 m on a side. Two
sides are held at 0o and two sides vary from 0 to
100o Find the steady state temperature.
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Boundary Conditions
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Mesh
4 3 2 1 j 0
i 0 1 2 3 4
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Mesh
y4 y3 y2
y1 y0
x0 x1 x2 x3 x4
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Mesh
y4 y3 y2
y1 y0
x0 x1 x2 x3 x4
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Notation
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PDE Second Order Central Difference Formulas
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PDE Second Order Central Difference Formulas
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Equations
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Solution of Equations

• For smaller systems use a direct method.
• Gaussian Elimination
• Matrix Factorization
• For larger systems use an iterative method.
• Gauss-Seidel
• Successive Over Relaxation

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Symmetric Block Tridiagonal Matrix Form
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Symmetric Block Tridiagonal Matrix Form
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Theorem
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Matrix Form
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Matrix Form
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Poissons Equation Problem
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PDE Second Order Central Difference Formulas
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PDE Second Order Central Difference Formulas
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Alternative Notation
N
W
E
S
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Other Topics

1. Alternative Solution Techniques.
2. Derivative B.C.
3. Nonuniform Boundaries.
4. Alternative Coordinate systems.

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Alternative Solution Procedure
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Alternating Direct Implicit Method
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Alternating Direct Implicit Method
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Example
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0
12
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Example
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Equations
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Gauss Elimination
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Gauss-Seidel
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Alternating Direct Implicit Method- Rows
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Alternating Direct Implicit Method- Columns
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Derivative B.C.

• Function value specified on boundary are called
  Dirichlet B.C.
• Derivative values specified on boundary are
  called Neumann Boundary Conditions

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Example
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Example
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Example
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Example
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Pseudo-Nodes
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Equations
4 equations in 6 unknowns.
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Derivative B.C
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Derivative B.C
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Irregular Boundary
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FDM Molecule
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Taylor Series
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FDM
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FDM
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FDM
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FDM
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FDM
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FDM
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FDM
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FDM