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Unformatted text preview: ACM 100c - Methods of Applied Mathematics Week 9 May 28, 2009 1 Greens functions for time dependent problems The Greens function approach can also be used for time dependent problems such as the wave or heat equation. In the following sections we will derive some important results. 1.1 Greens functions for the wave equation The derivation of the Greens function for the wave equation is considered first since it is a bit simpler than that for the heat equation. We start by looking at the inhomogeneous wave equation: ∂ 2 u ∂t 2 = c 2 ∇ 2 u + Q ( x , t ) . As we recall this equation requires initial conditions u ( x , 0) = f ( x ) u t ( x , 0) = g ( x ) as well as appropriate boundary conditions. The Greens function for the wave equation is defined similarly to that for Poisson’s equation: ∂ 2 G ∂t 2 = c 2 G ( x , x | t, t ) + δ ( x − x ) δ ( t − t ) . The boundary conditions for G are dictated by the type of boundary conditions for u on the boundary of the domain of interest. Recall for the Poisson equation, G = 0 if u satisfied inhomogeneous Dirichlet conditions for example. The Greens function has a physical interpretation - it is the response of the medium at some point x at some time t to an impulsive acceleration that was applied at x = x at time t = t . As a result of this interpretation we can see that G satisfies the constraint G ( x , t | x , t ) = 0 for t < t , because nothing can happen till the impulse turns on. For convenience we introduce the time difference τ = t − t as a convenient time variable. In this case our problem becomes ∂ 2 G ∂τ 2 = c 2 G + δ ( x − x ) δ ( τ ) , 1 and G = 0 τ < . Now recall Greens third formula. If we define L ≡ ∇ 2 then Greens third formula tells us that integraldisplayintegraldisplayintegraldisplay D [ u L v − v L u ] dV = integraldisplayintegraldisplay ∂D bracketleftbigg u ∂u ∂n − v ∂u ∂n bracketrightbigg dS. A similar type of result can be obtained for the wave equation but we have to redefine L as L = ∂ 2 ∂t 2 − c 2 ∇ 2 . For this “wave” operator we have that u L v − v L u = uv tt − vu tt − c 2 ( u ∇ 2 v − v ∇ 2 u ) . Using Greens third formula for ∇ 2 as well as the identity ∂ ∂t ( uv t − vu t ) = uv tt − vu tt we get that integraldisplay t f t i integraldisplayintegraldisplayintegraldisplay D [ u L v − v L u ] dV dt = integraldisplayintegraldisplayintegraldisplay D parenleftbigg u ∂v ∂t − v ∂u ∂t parenrightbigg dV vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle t f t i − c 2 integraldisplay t f t i integraldisplayintegraldisplay ∂D bracketleftbigg u ∂v ∂n − v ∂u ∂n bracketrightbigg dSdt....
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- Spring '09
- Boundary value problem, Partial differential equation, wave equation, greens, greens function