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

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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


Discriminant

Discriminant

PDE's


Classification

Classification

PDE's


Conic sections

Conic Sections

PDE's


Classification1

Classification

PDE's


Elliptical pde

Elliptical PDE

PDE's


Elliptical pde1

Elliptical PDE

PDE's


Fdm molecule

FDM Molecule

PDE's


Ode second order formula

ODE Second Order Formula

xi+1

xi

xi-1

PDE's


Ode second order central difference formulas

ODE Second Order Central Difference Formulas

PDE's


Ode second order central difference formulas1

ODE Second Order Central Difference Formulas

PDE's


Pde second order central difference formulas

PDE Second Order Central Difference Formulas

PDE's


Pde second order central difference formulas1

PDE Second Order Central Difference Formulas

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


Boundary conditions

Boundary Conditions

PDE's


Partial differential equations

Mesh

4

3

2

1

j = 0

i = 0 1 2 3 4

PDE's


Partial differential equations

Mesh

y4

y3

y2

y1

y0

x0 x1 x2 x3 x4

PDE's


Partial differential equations

Mesh

y4

y3

y2

y1

y0

x0 x1 x2 x3 x4

PDE's


Notation

Notation

PDE's


Pde second order central difference formulas2

PDE Second Order Central Difference Formulas

PDE's


Pde second order central difference formulas3

PDE Second Order Central Difference Formulas

PDE's


Equations

Equations:

PDE's


Partial differential 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


Symmetric block tridiagonal matrix form

Symmetric Block Tridiagonal Matrix Form

PDE's


Symmetric block tridiagonal matrix form1

Symmetric Block Tridiagonal Matrix Form

PDE's


Theorem

Theorem

PDE's


Matrix form

Matrix Form

PDE's


Matrix form1

Matrix Form

PDE's


Poisson s equation problem

Poisson’s Equation Problem

PDE's


Pde second order central difference formulas4

PDE Second Order Central Difference Formulas

PDE's


Pde second order central difference formulas5

PDE Second Order Central Difference Formulas

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


Alternative solution procedure

Alternative Solution Procedure

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


Gauss elimination

Gauss Elimination

PDE's


Gauss seidel

Gauss-Seidel

PDE's


Partial differential 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


Pseudo nodes

Pseudo-Nodes

PDE's


Equations2

Equations

4 equations in 6 unknowns.

PDE's


Derivative b c1

Derivative B.C

PDE's


Derivative b c2

Derivative B.C

PDE's


Irregular boundary

Irregular Boundary

PDE's


Fdm molecule1

FDM Molecule

PDE's


Taylor series

Taylor Series

PDE's


Partial differential equations

FDM

PDE's


Partial differential equations

FDM

PDE's


Partial differential equations

FDM

PDE's


Partial differential equations

FDM

PDE's


Partial differential equations

FDM

PDE's


Partial differential equations

FDM

PDE's


Partial differential equations

FDM

PDE's


Partial differential equations

FDM

PDE's


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