solh3 - Homework Set 3 Math 716 Due Wednesday February 7th...

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Homework Set 3: Math 716, Due: Wednesday, February 7th 1. Use energy method to prove uniqueness of classical solution to the initial value problem for the damped wave equation ( ǫ> 0): u tt + ǫu t = Δ u for x Ω R n , t> 0 , with u ( x , 0) = φ ( x ) , u t ( x , 0) = ψ ( x ) , u ( x ,t ) = 0 on Ω Solution: Define the Energy the same way as for the undamped ( ǫ = 0 case) wave equation: E ( t ) = 1 2 integraldisplay Ω bracketleftbig u 2 t + ( u ) 2 bracketrightbig dx Then, using equation, dE dt = integraldisplay Ω [ u t u tt + [ u ] · [ u t ]] dx = integraldisplay Ω bracketleftbig u t Δ u ǫu 2 t + [ u ] · [ u t ] bracketrightbig dx However, u t Δ u + [ u ] · [ u t ] = ∇ · [ u t u ] Assuming Ω is regular enough to apply divergence theorem, we obtain from the boundary condition u ( x,t ) = 0 on Ω (and hence u t ( x,t ) = 0 on Ω), we obtain integraldisplay Ω [ u t Δ u + [ u ] · [ u t ]] dx = integraldisplay Ω ∂u ∂n u t dx = 0 Therefore, dE dt = ǫ integraldisplay Ω u 2 t dx 0 Therefore, E ( t ) E (0) To prove uniqueness, consider two solutions u 1 , u 2 of the wave equation satisfying homo- geneous BC on Ω and same initial condition. Then u = u 1 u 1 will satisfy zero initial and boundary conditions. Hence from the above calculation E ( t ) E (0) = 1 2 integraldisplay Ω bracketleftbig u 2 t ( x, 0) + ( u ) 2 ( x, 0) bracketrightbig dx = 0 Therefore, E ( t ) = 0 implying from the expression for E that for any classical solution, u t ( x,t ) = 0 = u ( x,t ). This means u ( x,t ) = constant, independent of x and t . But since u ( x, 0) = 0, u ( x,t ) = 0, proving uniqueness of initial value problem. 2. Find representation of solution to heat equation with Neumann boundary conditions on a half-line u t = u xx 0 <x< , t > 0 , with u ( x, 0) = 0 , u x (0 ,t ) = sin t Solution: We note that v ( x,t ) = e - x sin t satisfies given Boundary condition, though not the PDE. Nonethless, define
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