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Unformatted text preview: 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 ( > 0): u tt + u t = u for x R n , t > , with u ( x , 0) = ( x ) , u t ( x , 0) = ( x ) , u ( x , t ) = 0 on Solution: Define the Energy the same way as for the undamped ( = 0 case) wave equation: E ( t ) = 1 2 integraldisplay bracketleftbig u 2 t + ( u ) 2 bracketrightbig dx Then, using equation, dE dt = integraldisplay [ u t u tt + [ u ] [ u t ]] dx = integraldisplay bracketleftbig u t u u 2 t + [ u ] [ u t ] bracketrightbig dx However, u t u + [ u ] [ u t ] = [ u t u ] Assuming is regular enough to apply divergence theorem, we obtain from the boundary condition u ( x, t ) = 0 on (and hence u t ( x, t ) = 0 on ), we obtain integraldisplay [ u t u + [ u ] [ u t ]] dx = integraldisplay u n u t dx = 0 Therefore, dE dt = integraldisplay u 2 t dx Therefore, E ( t ) E (0) To prove uniqueness, consider two solutions u 1 , u 2 of the wave equation satisfying homo- geneous BC on 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 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....
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- Spring '08