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D E M O N S T R A T I O MATHEMATICAVol. X X I XNo 41996Elzbieta Majewska, Jan PopiolekON THE SECOND BOUNDARY-VALUE PROBLEMFOR THE AIRY EQUATION1. IntroductionConsider the equationDzxu{x,t)-(1)Dtu{x,t)= 0.In [3] there has been examined the equation Dtu = mD^u which is calledthe Airy equation and is a linear version of the Korteweg-de Vries (KdV)equation. It arises in the description of the slow variation of a wave frontin coordinates moving with the wave. It also describes the propagation ofoscillatory wave packets. In [5], [6] it is proved that equation Dtu = D^u isone of the canonical forms of third order partial differential equations andit is called the equation with characteristics multiple (see [4], p. 132).The first boundary value problem (or also called the Cattabriga problem)for Airy equation has been examined in [2], [4]; moreover, in [3] the Cauchyproblem for this equation has been considered. Papers [9], [10] were devotedto solve contact problems for the said equation.This paper concerns the second boundary-value problem for equation(1) in the domainD = {(jc, i) G K 2 : 0 < X < 1, 0 < t < T},T = const. > 0.First, we shall examine properties of some integrals related to
Demonstratio Mathematica – de Gruyter
Published: Oct 1, 1996
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