Partial Differential Equations of Hyperbolic Type

Abstract

In the present chapter we consider the abstract Cauchy problem for differential equations of the hyperbolic type $$\upsilon ''\left( t \right) + A\upsilon \left( t \right) = f\left( t \right)\left( {0 \leqslant t \leqslant T} \right),\upsilon \left( 0 \right) = {{\upsilon }_{0}},\upsilon '\left( 0 \right) = {\text{ }}{{\upsilon '}_{0}}$$ in an arbitrary Hilbert space H with the self-adjoint positive definite operator A. The high order of accuracy two-step difference schemes generated by an exact difference scheme or by Taylor’s decomposition on three points for the numerical solutions of this problem are presented. Stability estimates for the solutions of these difference schemes are established. In applications, the stability estimates for solutions of a high order of accuracy difference schemes of the mixed type boundary-value problems for hyperbolic equations are obtained.