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# hw4 - u x y t is a solution of the heat equation u t = Δ u...

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Math 425/525, Spring 2011 Jerry L. Kazdan Problem Set 4 D UE : In class Thursday, Feb. 17 Late papers will be accepted until 1:00 PM Friday . 1. Solve the wave equation u tt = c 2 u xx for the semi-infinite string x 0 with the initial and boundary conditions u ( x , 0 ) = 3 - sin x , u t ( x , 0 ) = 0 , u ( 0 , t ) = 3 - t 2 . 2. [Weinberger p. 27 #3] Let u ( x , t ) be a solution of the inhomogeneous wave equation u tt - c 2 u xx = sin π x for 0 < x < 1, t > 0 with the boundary conditions u ( 0 , t ) = 0 and u ( 1 , t ) = 0. a) Find the solution if u ( x , 0 ) = 0 and u t ( x , 0 ) = 0. b) Find the solution if u ( x , 0 ) = x ( 1 - x ) and u t ( x , 0 ) = 0. 3. Let u ( x , t ) be the temperature at time t at the point x , - 1 x 1. Assume it satisfies the heat equation u t = u xx for 0 < t < with the boundary condition u ( - 1 , t ) = u ( 1 , t ) = 0 and initial condition u ( x , 0 ) = f ( x ) . a) Show that E ( t ) : = 1 2 Z 1 - 1 u 2 ( x , t ) dx is a decreasing function of t . b) Use this to prove uniqueness for the heat equation with these specified initial and boundary conditions. c) If u ( x , 0 ) = f ( x ) is an even function of x , show that the temperature u ( x , t ) at later times is also an even function of x Bonus Problems 1-B [Generalization of #3] Let u
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Unformatted text preview: u ( x , y , t ) is a solution of the heat equation u t = Δ u in a bounded region Ω ⊂ R 2 . Here one has the initial condition u ( x , y , ) = f ( x , y ) and boundary condition u ( x , y , t ) = 0 at points (x,y) on the boundary, ∂Ω and deﬁne E ( t ) : = 1 2 ZZ Ω u 2 ( x , y , t ) dxdy . a) Generalize #3(a)(b) to this setting. b) If Ω is symmetric under refection across the y-axis, so ( x , y ) → (-x , y ) and if the initial temperature is also symmetric, f ( x , y ) = f (-x , y ) , show that u ( x , y , t ) = u (-x , y , t ) for all points in Ω and all t > 0. REMARK: If Ω is the unit disk, { x 2 + y 2 < 1 } and f ( x , y ) depends only the distance to the origin, r : = p x 2 + y 2 , the same reasoning shows that the solution u also depends only on r . [Last revised: March 9, 2011] 1...
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