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Unformatted text preview: Lecture 35 Parabolic PDEs - Explicit Method Heat Flow and Diffusion In the previous sections we studied PDE that represent steady-state heat problem. There was no time variable in the equation. In this section we begin to study how to solve equations that involve time, i.e. we calculate temperature profiles that are changing. The conduction of heat and diffusion of a chemical happen to be modeled by the same differential equation. The reason for this is that they both involve similar processes. Heat conduction occurs when hot, fast moving molecules bump into slower molecules and transfer some of their energy. In a solid this involves moles of molecules all moving in different, nearly random ways, but the net effect is that the energy eventually spreads itself out over a larger region. The diffusion of a chemical in a gas or liquid simliarly involves large numbers of molecules moving in different, nearly random ways. These molecules eventually spread out over a larger region. In three dimensions, the equation that governs both of these processes is the heat/diffusion equation u t = c Δ u , where c is the coefficient of conduction or diffusion, and Δ u ( x, y, z ) = u xx + u yy + u zz . The symbol Δ in this context is called the Laplacian . If there is also a heat/chemical source, then it is incorporated a function g ( x, y, z, t ) in the equation as u t = c Δ u + g. In some problems the z dimension is irrelevent, either because the object in question is very thin, or u does not change in the z direction. In this case the equation is u t = c Δ u = c ( u xx + u yy ) ....
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- Fall '08
- Equations, Boundary value problem, Boundary conditions, explicit method