The numerical solution of some classes of the semilinear Sobolev-type equations

A unique solvability of the Cauchy problem for a class of semilinear Sobolev type equations of the second order is proved. The ideas and techniques, developed by G.A. Sviridyuk for the investigation of the Cauchy problem for a class of semilinear Sobolev type equations of the first order and by A.A. Zamyshlyaeva for the investigation of the high-order linear Sobolev type equations are used. We also used theory of the differential manifolds which was finally formed in S. Leng's works. In article we considered two cases. The first one is when an operator at the highest time derivative is continuously invertible. In this case for any point from a tangent bundle of an original Banach space there exists a unique solution lying in this space as trajectory. The second case when the operator isn't continuously invertible is of great interest for us. Hence we used the phase space method. Besides the Cauchy problem we considered the Showalter – Sidorov problem. The last generalizes the Cauchy problem and is more natural for Sobolev-type equation. In the last section described an algorithm of the numerical solution of Showalter – Sidorov problem for Sobolev-type equation of the second order.