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Boundary Value Problem

Hello to all.

I have the following BVP, consisting of the equations

eq1 = (1 - a^2)*D[f[x, y], {x, 2}] + D[f[x, y], {y, 2}] == 0;
eq2 = (1 - b^2)*D[g[x, y], {x, 2}] + D[g[x, y], {y, 2}] == 0;

and the boundary conditions

bc1 = (b^2 - 2)*D[f[x, y], {x, 2}] - 2*D[g[x, y], {x, 1}, {y, 1}] == (-
bc2 = (b^2 - 2)*D[g[x, y], {x, 2}] + 2*D[f[x, y], {x, 1}, {y, 1}] ==

where P a constant.

The problem describes the response of an elastic isotropic linear half
plane to the steady state motion of a concentrated normal load at the
surface. (The so called Cole-Huth problem).
a and b are the Mach numbers. Three cases can be distinguished:

a>b>1 (supersonic case)
a>1>b (transonic case)
1>a>b (subsonic case)

There a couple of solutions in the literature. One such is, integral
transform analysis. In fact, I have solved it with the aim of Laplace
transform. But I wonder if (and of course how!) Mathematica can be
utilized to solve above BVP.

Thank you very much in advance

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