Abstract
The initial-value problem for hyperbolic equation d2u(t)/dt2+A(t)u(t)=f(t)(0≤t≤T), u(0)=ϕ,u′(0)=ψ in a Hilbert space H with the self-adjoint positive definite operators A(t) is considered. The second order of accuracy difference scheme for the approximately solving this initial-value problem is presented. The stability estimates for the solution of this difference scheme are established.
Highlights
It is known that various mixed problems for the hyperbolic equations can be reduced to the initial-value problem d2u(t) dt2
A large cycle of works on difference schemes for hyperbolic partial differential equations, in which stability was established under the assumption that the magnitudes of the grid steps τ and h with respect to the time and space variables is connected
Of great interest is the study of absolute stable difference schemes of a high order of accuracy for hyperbolic partial differential equations, in which stability was established without any assumptions in respect of the grid steps τ and h
Summary
The initial-value problem for hyperbolic equation d2u(t)/dt2 + A(t)u(t) = f (t) (0 ≤ t ≤ T), u(0) = φ,u (0) = ψ in a Hilbert space H with the self-adjoint positive definite operators A(t) is considered. The second order of accuracy difference scheme for the approximately solving this initial-value problem is presented. The stability estimates for the solution of this difference scheme are established
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