5 - Properties of Solutions of Second Order PDE

5 - Properties of Solutions of Second Order PDE - MATH 220:...

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Unformatted text preview: MATH 220: PROPERTIES OF SOLUTIONS OF SECOND ORDER PDE ANDRAS VASY We have solved the initial value problem for the wave equation ( 2 t c 2 2 x ) u = 0 , u ( x, 0) = ( x ) , u t ( x, 0) = ( x ) , namely we showed that the solution is u ( x, t ) = 1 2 ( ( x + ct ) + ( x ct )) + 1 2 c integraldisplay x + ct x ct ( ) d. There are a few facts that can be read off from this expression immediately. We consider t > 0 here; t < 0 is similar. First, for t > 0, u ( x , t ) depends on the initial data just at the two points x ct , while it depends on the values of in the whole interval [ x ct , x + ct ]. Thus, we call the interval [ x ct , x + ct ] the domain of dependence of ( x , t ): if the initial conditions vanish there, the solution vanishes at ( x , t ). Note that the straight lines x ct = x ct and x + ct = x + ct which go through ( x , t ) and ( x ct , 0) are characteristics. In fact, it is convenient (for reasons that will be more clear when we solve the inhomogeneous wave equation, square u = f ) to consider the domain of dependence of ( x, t ) to be the whole region D x ,t = { ( x, t ) : t t , | x x | c ( t t ) } . This is the backward characteristic triangle from ( x , t ): its sides are the charac- teristics x ct = x ct and x + ct = x + ct . With this definition, if the initial data are imposed at t = T instead, where T < t , then the solution u at ( x , t ) depends on the initial data in the interval [ x c ( t T ) , x + c ( t T )], which is just the segment in which the line t = T intersects the backward characteristic triangle. Indeed, a change of variables (replacing t by = t T ) shows that the solution of ( 2 t c 2 2 x ) u = 0 , u ( x, T ) = T ( x ) , u t ( x, T ) = T ( x ) , is u ( x, t ) = 1 2 ( T ( x + c ( t T )) + T ( x c ( t T ))) + 1 2 c integraldisplay x + c ( t T ) x c ( t T ) T ( ) d. On the flipside, we may ask at which points ( x , t ) does, or does not, the initial data at ( x, 0) influence the solution. More precisely, we can ask at which points ( x , t ) is the solution guaranteed to be unaffected if we change the initial data at ( x, 0)? This happens exactly if ( x, 0) is not in the backward characteristic triangle from ( x , t ), D x ,t , i.e. (keeping in mind t > 0) if | x x | > ct . Conversely, if | x x | ct , then the solution may change (and in general does change) if and are changed at ( x, 0). Thus, we call the forward characteristic triangle D + x, = { ( x , t ) : t , | x x | ct } 1 2 ANDRAS VASY the domain of influence of ( x, 0). More generally, for initial data at t = T , the (forward) domain of influence of ( x, T ) is D + x,T = { ( x , t ) : t T, | x x | c ( t T ) } ....
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This note was uploaded on 03/16/2010 for the course CME 303 taught by Professor Vasy during the Fall '10 term at Stanford.

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5 - Properties of Solutions of Second Order PDE - MATH 220:...

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