This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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....
View Full
Document
 Spring '08
 Tanveer,S
 Math

Click to edit the document details