Partial Differential Equations

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Presentation transcript:

Partial Differential Equations Chapter 12 Burden and Faires 11/18/2018 PDE's

General 2nd Order Form 11/18/2018 PDE's

Discriminant 11/18/2018 PDE's

Classification 11/18/2018 PDE's

Conic Sections 11/18/2018 PDE's

Classification 11/18/2018 PDE's

Elliptical PDE 11/18/2018 PDE's

Elliptical PDE 11/18/2018 PDE's

FDM Molecule 11/18/2018 PDE's

ODE Second Order Formula xi-1 xi xi+1 11/18/2018 PDE's

ODE Second Order Central Difference Formulas 11/18/2018 PDE's

ODE Second Order Central Difference Formulas 11/18/2018 PDE's

PDE Second Order Central Difference Formulas 11/18/2018 PDE's

PDE Second Order Central Difference Formulas 11/18/2018 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. 11/18/2018 PDE's

Boundary Conditions 11/18/2018 PDE's

Mesh 4 3 2 1 j = 0 i = 0 1 2 3 4 11/18/2018 PDE's

Mesh y4 y3 y2 y1 y0 x0 x1 x2 x3 x4 11/18/2018 PDE's

Mesh y4 y3 y2 y1 y0 x0 x1 x2 x3 x4 11/18/2018 PDE's

Notation 11/18/2018 PDE's

PDE Second Order Central Difference Formulas 11/18/2018 PDE's

PDE Second Order Central Difference Formulas 11/18/2018 PDE's

Equations: 11/18/2018 PDE's

11/18/2018 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 11/18/2018 PDE's

Symmetric Block Tridiagonal Matrix Form 11/18/2018 PDE's

Symmetric Block Tridiagonal Matrix Form 11/18/2018 PDE's

Theorem 11/18/2018 PDE's

Matrix Form 11/18/2018 PDE's

Matrix Form 11/18/2018 PDE's

Poisson’s Equation Problem 11/18/2018 PDE's

PDE Second Order Central Difference Formulas 11/18/2018 PDE's

PDE Second Order Central Difference Formulas 11/18/2018 PDE's

Alternative Notation N W E S 11/18/2018 PDE's

Other Topics Alternative Solution Techniques. Derivative B.C. Nonuniform Boundaries. Alternative Coordinate systems. 11/18/2018 PDE's

Alternative Solution Procedure 11/18/2018 PDE's

Alternating Direct Implicit Method 11/18/2018 PDE's

Alternating Direct Implicit Method 11/18/2018 PDE's

Example 12 12 11/18/2018 PDE's

Example 11/18/2018 PDE's

Equations 11/18/2018 PDE's

Gauss Elimination 11/18/2018 PDE's

Gauss-Seidel 11/18/2018 PDE's

11/18/2018 PDE's

Alternating Direct Implicit Method - Rows 11/18/2018 PDE's

Alternating Direct Implicit Method - Columns 11/18/2018 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 11/18/2018 PDE's

Example: 11/18/2018 PDE's

Example 11/18/2018 PDE's

Example 11/18/2018 PDE's

Example 11/18/2018 PDE's

Pseudo-Nodes 11/18/2018 PDE's

Equations 4 equations in 6 unknowns. 11/18/2018 PDE's

Derivative B.C 11/18/2018 PDE's

Derivative B.C 11/18/2018 PDE's

Irregular Boundary 11/18/2018 PDE's

FDM Molecule 11/18/2018 PDE's

Taylor Series 11/18/2018 PDE's

FDM 11/18/2018 PDE's

FDM 11/18/2018 PDE's

FDM 11/18/2018 PDE's

FDM 11/18/2018 PDE's

FDM 11/18/2018 PDE's

FDM 11/18/2018 PDE's

FDM 11/18/2018 PDE's

FDM 11/18/2018 PDE's