Decomposition schemes for abstract hyberbolic equation

Author: Ketevan Gomiashvili
Keywords: Linear abstract hyperbolic equation, Quasi-linear abstract hyperbolic equation, homogeneus abstract hyperbolic equation, parallel decomposition schemes, Estimation of error
Annotation:

In the present work, we consider the Cauchy problem for abstract hyperbolic equation in the Hilbert space H. Operator A from ellipse part of equation, is sum of A_1,A_2, . . . ,A_m self-adjoint , positively defined operators. The parallel type of decomposition scheme for the approximate solution of given problem is constructed. The idea of this scheme is that in every local interval according to operators A_1,A_2, . . . ,A_m the classical discrete analogues are solved independently. The weighted average deducted from the solutions is called as approximate solution at the right end of the local interval. The convergence of offered scheme is proved. Error of approximate salution and diskrete analogue of first order derivative are estimated, when initial data satisfies some natural conditions for existence of the salution. In the last part, we consider the high order decomposition scheme for the Cauchy problem for homogeneus abstract hyperbolic equation. The high order schemes of parallel decomposition are constructed based on rational approximation for cosine operator function. We use the methods already considered in first chapters to estimate error of approximate salution.

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