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Partial Differential Equations

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Partial Differential Equations

Chapter 12

Burden and Faires

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xi+1

xi

xi-1

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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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4

3

2

1

j = 0

i = 0 1 2 3 4

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y4

y3

y2

y1

y0

x0 x1 x2 x3 x4

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y4

y3

y2

y1

y0

x0 x1 x2 x3 x4

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- 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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N

W

E

S

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- Alternative Solution Techniques.
- Derivative B.C.
- Nonuniform Boundaries.
- Alternative Coordinate systems.

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12

0

12

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

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4 equations in 6 unknowns.

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