A note on the difference schemes for hyperbolic‐elliptic equations

Abstract

The nonlocal boundary value problem for hyperbolic-elliptic equationd2u(t)/dt2+Au(t)=f(t),(0≤t≤1),−d2u(t)/dt2+Au(t)=g(t),(−1≤t≤0),u(0)=ϕ,u(1)=u(−1)in a Hilbert spaceHis considered. The second order of accuracy difference schemes for approximate solutions of this boundary value problem are presented. The stability estimates for the solution of these difference schemes are established.

Second Order Equations in Functional Spaces: Qualitative and Discrete Well-Posedness

“…Section 7.3 is based on results of [70][71][72][73][74][75][76]. Section 7.4 is based on results of [77][78][79][80][81][82][83][84][85][86][87][88][89][90][91]. Section 7.5 is devoted to stochastic hyperbolic equations.…”

“…The generalization of stability estimates results of [77,126] was presented in [86] for the solution of the multipoint nonlocal boundary value problem…”

“…In [77], applying the second order of accuracy difference scheme for hyperbolic equations and elliptic equations and the second order approximation formulae for nonlocal condition and continuity condition at = 0, the following second order of accuracy difference schemes…”

The hyperbolic-elliptic equation with the nonlocal condition

“…They also play a very important role for mathematical modelling in many branches of science, engineering and industry. Theory and numerical methods of solutions of the boundary value problems endowed with the local and nonlocal boundary conditions for partial differential equations have been investigated by many researchers ( [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] and the references therein).…”

“…The element u.t/ belongs to D.A/ for all t 2 OE 1, 1, and the function Au.t/ is continuous on OE 1, 1. 3. u.t/ satisfies the equations and boundary conditions (1).…”