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Partial Differential EquationsPowerPoint Presentation

Partial Differential Equations

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Presentation Transcript

General 2nd Order Form

PDE's

Discriminant

PDE's

Classification

PDE's

Conic Sections

PDE's

Classification

PDE's

Elliptical PDE

PDE's

Elliptical PDE

PDE's

FDM Molecule

PDE's

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.

PDE's

Boundary Conditions

PDE's

Notation

PDE's

Equations:

PDE's

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

PDE's

Theorem

PDE's

Matrix Form

PDE's

Matrix Form

PDE's

Other Topics

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

PDE's

Alternating Direct Implicit Method

PDE's

Alternating Direct Implicit Method

PDE's

Example

PDE's

Equations

PDE's

Gauss Elimination

PDE's

Gauss-Seidel

PDE's

Alternating Direct Implicit Method- Rows

PDE's

Alternating Direct Implicit Method- Columns

PDE's

Derivative B.C.

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

PDE's

Example:

PDE's

Example

PDE's

Example

PDE's

Example

PDE's

Pseudo-Nodes

PDE's

Derivative B.C

PDE's

Derivative B.C

PDE's

Irregular Boundary

PDE's

FDM Molecule

PDE's

Taylor Series

PDE's

FDM

PDE's

FDM

PDE's

FDM

PDE's

FDM

PDE's

FDM

PDE's

FDM

PDE's

FDM

PDE's

FDM

PDE's

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