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5 - Properties of Solutions of Second Order PDE

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

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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) = φ ( x ) , u t ( x, 0) = ψ ( x ) , namely we showed that the solution is u ( x,t ) = 1 2 ( φ ( x + ct ) + φ ( x ct )) + 1 2 c integraldisplay x + ct x ct ψ ( σ ) dσ. 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 > 0, u ( x 0 ,t 0 ) depends on the initial data φ just at the two points x 0 ± ct 0 , while it depends on the values of ψ in the whole interval [ x 0 ct 0 ,x 0 + ct 0 ]. Thus, we call the interval [ x 0 ct 0 ,x 0 + ct 0 ] the domain of dependence of ( x 0 ,t 0 ): if the initial conditions vanish there, the solution vanishes at ( x 0 ,t 0 ). Note that the straight lines x ct = x 0 ct 0 and x + ct = x 0 + ct 0 which go through ( x 0 ,t 0 ) and ( x 0 ± ct 0 , 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 0 ,t 0 = { ( x,t ) : t t 0 , | x x 0 | ≤ c ( t 0 t ) } . This is the backward characteristic triangle from ( x 0 ,t 0 ): its sides are the charac- teristics x ct = x 0 ct 0 and x + ct = x 0 + ct 0 . With this definition, if the initial data are imposed at t = T instead, where T <t 0 , then the solution u at ( x 0 ,t 0 ) depends on the initial data in the interval [ x 0 c ( t 0 T ) ,x 0 + c ( t 0 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 τ = t T ) shows that the solution of ( 2 t c 2 2 x ) u = 0 , u ( x,T ) = φ T ( x ) , u t ( x,T ) = ψ T ( x ) , is u ( x,t ) = 1 2 ( φ T ( x + c ( t T )) + φ T ( x c ( t T ))) + 1 2 c integraldisplay x + c ( t T ) x c ( t T ) ψ T ( σ ) dσ. On the flipside, we may ask at which points ( x 0 ,t 0 ) does, or does not, the initial data at ( x, 0) influence the solution. More precisely, we can ask at which points ( x 0 ,t 0 ) 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 0 ,t 0 ), D x 0 ,t 0 , i.e. (keeping in mind t 0 > 0) if | x x 0 | >ct 0 . Conversely, if | x x 0 | ≤ ct 0 , then the solution may change (and in general does change) if φ and ψ are changed at ( x, 0). Thus, we call the forward characteristic triangle D + x, 0 = { ( x 0 ,t 0 ) : t 0 0 , | x x 0 | ≤ ct 0 } 1

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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 0 ,t 0 ) : t 0 T, | x x 0 | ≤ c ( t 0 T ) } . That the size of the intersection of the domain of influence with the line t = t 0 increases at speed c is called Huygens’ principle , and is also valid for wave equations with variable coefficients: waves propagate (at most) as fast as c , so c is reasonably called the speed of waves. Although the solution at ( x 0 ,t 0 ) depends on the initial data everywhere inside its domain of dependence, its dependence on the data may not be very significant. One way to think about this is the following. ‘Information’ is carried by singularities of solutions to the PDE, e.g. one flips a switch, and gets a jump in the amplitude of
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5 - Properties of Solutions of Second Order PDE - MATH 220...

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