Partial differential equations
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Partial Differential Equations PowerPoint PPT Presentation


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Partial Differential Equations. Chapter 12 Burden and Faires. General 2 nd Order Form. Discriminant. Classification. Conic Sections. Classification. Elliptical PDE. Elliptical PDE. FDM Molecule. ODE Second Order Formula. x i+1. x i. x i-1.

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Partial Differential Equations

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Partial Differential Equations

Chapter 12

Burden and Faires

PDE's


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


ODE Second Order Formula

xi+1

xi

xi-1

PDE's


ODE Second Order Central Difference Formulas

PDE's


ODE Second Order Central Difference Formulas

PDE's


PDE Second Order Central Difference Formulas

PDE's


PDE Second Order Central Difference Formulas

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


Mesh

4

3

2

1

j = 0

i = 0 1 2 3 4

PDE's


Mesh

y4

y3

y2

y1

y0

x0 x1 x2 x3 x4

PDE's


Mesh

y4

y3

y2

y1

y0

x0 x1 x2 x3 x4

PDE's


Notation

PDE's


PDE Second Order Central Difference Formulas

PDE's


PDE Second Order Central Difference Formulas

PDE's


Equations:

PDE's


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


Symmetric Block Tridiagonal Matrix Form

PDE's


Symmetric Block Tridiagonal Matrix Form

PDE's


Theorem

PDE's


Matrix Form

PDE's


Matrix Form

PDE's


Poisson’s Equation Problem

PDE's


PDE Second Order Central Difference Formulas

PDE's


PDE Second Order Central Difference Formulas

PDE's


Alternative Notation

N

W

E

S

PDE's


Other Topics

  • Alternative Solution Techniques.

  • Derivative B.C.

  • Nonuniform Boundaries.

  • Alternative Coordinate systems.

PDE's


Alternative Solution Procedure

PDE's


Alternating Direct Implicit Method

PDE's


Alternating Direct Implicit Method

PDE's


Example

12

0

12

PDE's


Example

PDE's


Equations

PDE's


Gauss Elimination

PDE's


Gauss-Seidel

PDE's


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


Equations

4 equations in 6 unknowns.

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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