16. Conclusion

The constructed singular boundary integral equations in the supersonic case are not classical equations because the solution inside a domain is determined by the boundary values of stresses and displacement rates rather than displacements themselves, unlike the Somigliana formula. In addition, the domain of integration over a boundary surface substantially depends on z, which is specific for hyperbolic equations. This complicates finding solutions of such problems by the successive approximation method. However, for the numerical discretization of singular boundary integral equations, the method of boundary elements makes it possible to use standard methods of computational mathematics for a computer implementation of the solution of such problems. The aforementioned boundary value problems model the dynamics of underground structures like transport tunnels and extended excavations subjected to the dynamic influence of moving vehicles and seismic loads. They permit one to study the dynamics of a rock mass in a neighborhood of underground structures depending on its physicalmechanical properties, the velocity of moving transport, specific features of the transport load, and the geometric properties of structures in technical computations of displacements and the stress-strain state of the mass away from the tunnel.
