Partial differential equations
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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

Partial Differential Equations

Chapter 12

Burden and Faires

PDE's


General 2 nd order form
General 2nd Order Form

PDE's









Ode second order formula
ODE Second Order Formula

xi+1

xi

xi-1

PDE's






Example problem
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



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
Notation

PDE's




Equations
Equations:

PDE's



Solution of equations
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
Theorem

PDE's







Alternative notation
Alternative Notation

N

W

E

S

PDE's


Other topics
Other Topics

  • Alternative Solution Techniques.

  • Derivative B.C.

  • Nonuniform Boundaries.

  • Alternative Coordinate systems.

PDE's



Alternating direct implicit method
Alternating Direct Implicit Method

PDE's


Alternating direct implicit method1
Alternating Direct Implicit Method

PDE's


Example
Example

12

0

12

PDE's


Example1
Example

PDE's


Equations1
Equations

PDE's





Alternating direct implicit method rows
Alternating Direct Implicit Method- Rows

PDE's


Alternating direct implicit method columns
Alternating Direct Implicit Method- Columns

PDE's


Derivative b c
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


Example2
Example:

PDE's


Example3
Example

PDE's


Example4
Example

PDE's


Example5
Example

PDE's



Equations2
Equations

4 equations in 6 unknowns.

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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