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Unformatted text preview: Notes on the diffusion equation Patrick De Leenheer * March 27, 2009 In previous notes we considered the diffusion equation on R , and obtained the fundamental so- lution. Here we study the equation on a finite interval [0 ,l ] under two types of boundary conditions, namely Dirichlet and Neumann boundary conditions. Dirichlet Let u t = Du xx , u (0 ,t ) = u ( l,t ) = 0 for all t > . (1) We will not specify an initial condition, and only attempt to find nonzero solutions with a particular form using the method of separation of variables : u ( x,t ) = X ( x ) T ( t ) . Plugging this into the diffusion equation we find that T DT = X 00 X =- λ (2) for some constant λ which is yet to be determined. Then T ( t ) = T (0)e- Dλt , and since we are not interested in trivial solutions we assume that T (0) 6 = 0. Then u (0 ,t ) = u ( l,t ) = 0 implies that X (0) = X ( l ) = 0 . This leads to the boundary value problem (BVP) : X 00 + λX = 0 , X (0) = X ( l ) = 0 . There are 3 cases to consider: λ < , = 0 ,> 0. Case 1 : λ < 0. By standard results of linear 2nd order ODE’s we first find the general solution: X ( x ) = c 1 e √- λx + c 2 e- √- λx . The BC X (0) = X ( l ) = 0 then imply that c 1 = c 2 = 0, so we find a trivial solution which must be discarded. In other words, λ cannot be negative....
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This note was uploaded on 09/12/2011 for the course MAP 4305 taught by Professor Deleenheer during the Summer '06 term at University of Florida.
- Summer '06