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Unformatted text preview: ACM 100c - Methods of Applied Mathematics Week 7 May 9, 2009 1 The Laplace and Poisson equations In this section we will turn our attention to the late time behavior of the heat equation but in 2 and 3 space dimensions. We are thus interested in ∂u ∂t = a 2 ∂ 2 u ∂x 2 + ∂ 2 u ∂y 2 + f ( x,y ) in two space dimensions and ∂u ∂t = a 2 ∂ 2 u ∂x 2 + ∂ 2 u ∂y 2 + ∂ 2 u ∂z 2 + f ( x,y,z ) in three space dimensions. More generally, if we want to be free of particular coordinate systems then ∂u ∂t = a 2 ∇ 2 u + f ( r ,t ) , where r represents some vector in 2 or 3 dimensional space. As we have seen, given some initial conditions, time independent boundary conditions and time independent forcing functions, f ( x,y,z ) , we know that the generic behavior of solutions of this equation is that the solution u ( x,y,z,t ) will tend to an equilibrium solution assuming the boundary conditions and the forcing are properly compatible to allow the existence of an equilibrium solution. In that case, as t → ∞ , the solution u ( x,y,z,t ) becomes time independent and satisfies ∇ 2 u = f ( x,y,z ) /a 2 . If the boundary conditions are Dirichlet conditions, then this solution will always exist. If the boundary conditions are Neumann conditions, the solution will only exist if the right hand side is compatible with the boundary conditions and in that case the initial condition plays a role in making the solution unique. The equation ∇ 2 u = f ( x,y,z ) /a 2 is called Poisson’s equation and if f = 0 , then it is called Laplace’s equation . We will see that the solutions of these equations have their own unique properties that are characteristic of these equations. 1 2 Mathematical context for Poisson’s equation The Poisson equation differs from the other equations we have studied so far in that it is not an initial value problem. Given that it represents the equilibrium solution for either the heat or wave equation, we can see that the intuitive required conditions for a solution are just boundary conditions. From a mathematical point of view, the equation also differs in a structural way from other PDE’s. To date, given one time and one space dimension, we have studied the heat equation: u t = ku xx , the wave equation u tt = c 2 u xx , and we now consider the Laplace or Poisson equation which looks like u xx =- u yy . So we see that the Laplace equation features two derivatives with respect to two variables as does the wave equation but the signs differ. This will be shown to have a major effect on the nature of the solutions. 3 The concept of well-posedness A PDE and its boundary and initial conditions (generally referred to as the data of the PDE), constitute what is called a well-posed problem if the following conditions are met: 1. the solution exists 2. the solution is unique 3. the solution is smooth with respect to perturbations in the data....
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- Spring '09
- Boundary value problem, Boundary conditions, Green's function