Introduction to PDE classification Numerical Methods for PDEs Spring 2007 Jim E. Jones References: Partial Differential Equations of Applied Mathematics, Zauderer Wikopedia, Partial Differential Equation

PDE classified by discriminant: b 2 -4ac. –Negative discriminant = Elliptic PDE. Example Laplace’s equation –Zero discriminant = Parabolic PDE. Example Heat equation –Positive discriminant = Hyperbolic PDE. Example Wave equation Partial Differential Equations (PDEs) : 2 nd order model problems

Example: Parabolic Equation (Finite Domain) Heat equation Typical Boundary Conditions x=0 x=L/2 x=-L/2

Example: Parabolic Equation Heat equation Typical Boundary Conditions x=0 x=L/2 x=-L/2 Initial temperature profile in rod Temperatures for end of rod

Example: Parabolic Equation (Infinite Domain) Heat equation Dirac Delta Boundary Conditions x=0

Dirac Delta Function The Dirac delta function is the limit of Physically it corresponds to a localized intense source of heat

Example: Parabolic Equation (Infinite Domain) Heat equation Dirac Delta Boundary Conditions Solution (verify)

Example: Parabolic Equation (Infinite Domain) t=.01 t=.1 t=1t=10

Typically describe time evolution towards a steady state. –Model Problem: Describe the temperature evolution of a rod whose ends are held at a constant temperatures. Initial conditions have immediate, global effect –Point source at x=0 makes temperature nonzero throughout domain for all t > 0. Parabolic PDES

Example: Hyperbolic Equation (Infinite Domain) Heat equation Boundary Conditions

Example: Hyperbolic Equation (Infinite Domain) Heat equation Boundary Conditions Solution (verify)

Hyperbolic Equation: characteristic curves x-ct=constant x+ct=constant x t (x,t)

Example: Hyperbolic Equation (Infinite Domain) x-ct=constant x+ct=constant x t (x,t) The point (x,t) is influenced only by initial conditions bounded by characteristic curves.

Example: Hyperbolic Equation (Infinite Domain) Heat equation Boundary Conditions

Example: Hyperbolic Equation (Infinite Domain) t=.01t=.1 t=1t=10

Typically describe time evolution with no steady state. –Model problem: Describe the time evolution of the wave produced by plucking a string. Initial conditions have only local effect –The constant c determines the speed of wave propagation. Hyperbolic PDES

Example: Elliptic Equation (Finite Domain) Laplace’s equation Typical Boundary Conditions 

PDE solution (verify) The Problem

Elliptic Solution

Typically describe steady state behavior. –Model problem: Describe the final temperature profile in a plate whose boundaries are held at constant temperatures. Boundary conditions have global effect Elliptic PDES

PDE classified by discriminant: b 2 -4ac. –Negative discriminant = Elliptic PDE. Example Laplace’s equation –Zero discriminant = Parabolic PDE. Example Heat equation –Positive discriminant = Hyperbolic PDE. Example Wave equation Partial Differential Equations (PDEs) : 2 nd order model problems