A note on the second order of accuracy difference schemes for hyperbolic-parabolic equations

The nonlocal boundary value problem for hyperbolic-parabolic equationd2u(t)dt2+Au(t)=f(t)(0⩽t⩽1),du(t)dt+Au(t)=g(t)(-1⩽t⩽0),u(-1)=αu(μ)+ϕ,0⩽α⩽1,0<μ⩽1in a Hilbert space H is considered. The second order of accuracy difference schemes approximately solving this boundary value problem are presented. The stability estimates for the solution of these difference schemes are established. In applications, the stability estimates for the solutions of the difference schemes of the mixed type boundary value problems for hyperbolic-parabolic equations are obtained. The theoretical statements for the solution of these difference schemes for hyperbolic-parabolic equation are supported by the results of numerical experiments.