An application of the contour integral method to a mixed problem of dynamic impact theory (Q1874853)

We study the problem on transverse vibrations of a thick viscoelastic plate induced by a dynamic impact at a single point (an instantaneous application of a lumped force). In the mathematical statement, the problem is reduced to finding the function \(u(x,y,t)\) that describes the displacement of points of the midplane of the plate. This function must satisfy the equation \[ {\partial^2u \over\partial t^2}+ \alpha^2\Delta \Delta u+\beta^2 {\partial\over \partial t}(\Delta\Delta u)= {A\over\rho h}\delta (x-x_0,y-y_0) \] in the domain \(D_0=(0<x<a\), \(0<y<b\), \(0<t\leq T<\infty)\), with initial and boundary conditions.