Asymptotic Methods for the Solu of Dispersive Hyperbolic Equations
Asymptotic Methods for the Solu of Dispersive Hyperbolic Equations
Robert M Lewis
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Zja)[i-t hj b;;]-V2. 1-t X s", 1/2 00 O v» ' l. Th "s" (h5) (1*6) and Summarizing our results, we see that u * u + u where u^». -z^(0)[i-th^ ^o]~^^^ ^"^ { ^'^^% + -^ t) j, X - -a-gt u_^ z^(a)[l-th^ 3^""^^^ ^^ f ^^^% +it) I, X = a-tgt. (^7) (^) This con5)letes the solutlcm of the initial-boundary value problem. From (47) and (1*8) we note that u^(t, x) = -u_(t, -x). (k9) Of course {k9) is easily obtained by the "image method** which yields the solution of the InitiaX-boundary Value problem at on...ce in terms of the solirtion u of the initial value problem. Thus this method provides a check on the construction we have vised. The above problem, though essentially trivial suffices to * In this case the image method would consist in extending tlB initial data to X
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