PDEs

We shall spend most of the rest of the semester looking at computational methods for partial differential equations. We begin with a brief refresher.

Consider a function u=u(x,y) defined in the plane. A PDE relates changes in u as both x and y change. Classical equations arise in problems of physics and engineering.

• Laplace's equation is an elliptic PDE, describing, for instance, the time-independent distribution of temperature in a plate that is heated (or insulated) on its edges. The equation is

  

  A specific example is on the boundary, where the region under consideration is the unit square . The solution is . An alternative to the Dirichlet boundary conditions (u specified) is Neumann boundary considerations (normal derivative specified). For example, satisfies the equation on .

  The boundary value ODEs we studied are 1D examples of behavior of this type.

• The Heat equation is a parabolic equation, and describes the time-dependent distribution of temperature. We shall use the variables x, t instead of x, y. The heat equation is

  

  Note here that we must specify the initial condition u(x,0) as well as the boundary conditions . It should be clear that the heat equation reduces to the Laplace equation in the steady-state.

• The Wave equation is a hyperbolic PDE, and describes the propagation of sound waves in a medium. The equation is

  

  Here we have to specify the initial values of both u and , and boundary conditions.

  The wave equation can be written as a system. Consider Then

  

  In addition, appropriate combinations of v, w must be specified at the bondaries.

• Elliptic equations
• Hyperbolic equations
• Parabolic equations