ME 523 Numerical Solutions Applied To Heat Transfer And Fluid Mechanics Problems

Classification of partial differential equations

The contents of this page are simply a review of a classification method that you should have seen in an undergraduate PDE class. The lecture will focus more on material in your first reading assignment (see the ANGEL Lessons tab) from an Oak Ridge National Laboratory web site, providing more general techniques for classifying PDEs. Read sections 4 and 5 of this material carefully.

Boundary Value Problems (Equilibrium Problems)

Mathematical Model (valid in region R) : Conservation Equation
Boundary conditions (Neumann and/or Dirichlet Classification: Elliptic Equations

Applications:

Steady State Heat Conduction
Invicid Flow (Potential)
Viscous Flow Initial Value Problems (Propagation Problems) Mathematical Model (valid in region R, with possible moving boundary) : Conservation Equation
Boundary conditions
Initial Conditions Classification: Parabolic, Hyperbolic

Applications:

Transient Pressure Waves
Propagation of Momentum
Propagation of Stresses How can we classify a mathematical model?

Look at the conservation equation.

Quasi-Linear second order PDE

where a, b, c, and f can be constants or functions of

b² – 4 ac < 0	Equation is Elliptic
b² – 4 ac = 0	Equation is Parabolic
b² – 4 ac > 0	Equation is Hyperbolic

Example: Laplace or Poisson Equation (x, y)

Laplace

Poisson

Classification a = 1, b = 0, c = 1 b² – 4 ac = 0 – 4 (1) (1) = -4 Equations are Elliptic Example: Fourier Equation

Heat Conduction

We have 2 space variables and one time variable. Neglect one space variable since both space variables are of the same form. (Second Derivatives)

a = 0, b = 0, c = k b² – 4 ac = 0 – 4 k 0 = 0 Equation is Parabolic Example: Navier-Stokes Equation - Steady State and Incompressible Flow Variables: u(x, y), v(x, y), p(x, y)

Continuity Equation: