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Unformatted text preview: π, t ) = 0 , and: v t = v xx , v (0 , t ) = v ( π, t ) = 0 . Suppose that at the initial time t = 0 one has the inequality: v ( x, 0) 6 u ( x, 0) . Show that for all positive times 0 < t one has that: v ( x, t ) 6 u ( x, t ) . Please explain carefully your answer. (Hint: Consider the function w = u-v .) b) Notice that the function: u ( x, t ) = e-t sin( x ) , solves the above heat equation with zero Dirichlet boundary conditions on the interval [0 , π ]. Show that if v ( x, t ) is any solution to the (same) heat equation boundary value problem: v t = v xx , v (0 , t ) = v ( π, t ) = 0 , with the additional property that at t = 0 (on [0 , π ]):-sin( x ) 6 v ( x, t ) 6 sin( x ) , then one always has: | v ( x, t ) | 6 e-t . (Hint: Recall that if-u 6 v 6 u , with 0 6 u , then one also has | v | 6 u .)...
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- Fall '09
- Boundary value problem, Dirichlet boundary conditions