A note on the difference schemes for hyperbolic-elliptic equations

The nonlocal boundary value problem for hyperbolic-elliptic equation d2u(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 space H is 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

On well-posedness of the nonlocal boundary value problem for parabolic difference equations

We consider the nonlocal boundary value problem for difference equations (uk−uk−1)/τ+Auk=φk, 1≤k≤N, Nτ=1, and u0=u[λ/τ]+φ, 0<λ≤1, in an arbitrary Banach space E with the strongly positive operator A. The well-posedness of this nonlocal boundary value problem for difference equations in various Banach spaces is studied. In applications, the stability and coercive stability estimates in Hölder norms for the solutions of the difference scheme of the mixed-type boundary value problems for the parabolic equations are obtained. Some results of numerical experiments are given

A note on the difference schemes for hyperbolic equations

The initial value problem for hyperbolic equations d 2u(t)/dt 2+A u(t)=f(t)(0≤t≤1),u(0)=φ,u′(0)=ψ, in a Hilbert space H is considered. The first and second order accuracy difference schemes generated by the integer power of A approximately solving this initial value problem are presented. The stability estimates for the solution of these difference schemes are obtained

On the stability of the linear delay differential and difference equations

We consider the initial-value problem for linear delay partial differential equations of the parabolic type. We give a sufficient condition for the stability of the solution of this initial-value problem. We present the stability estimates for the solutions of the first and second order accuracy difference schemes for approximately solving this initial-value problem. We obtain the stability estimates in Hölder norms for the solutions of the initial-value problem of the delay differential and difference equations of the parabolic type

Coercive solvability of the nonlocal boundary value problem for parabolic differential equations

The nonlocal boundary value problem, v′(t)+Av(t)=f(t)(0≤t≤1),v(0)=v(λ)+μ(0<λ≤1), in an arbitrary Banach space E with the strongly positive operator A, is considered. The coercive stability estimates in Hölder norms for the solution of this problem are proved. The exact Schauder's estimates in Hölder norms of solutions of the boundary value problem on the range {0≤t≤1,xℝ n} for 2m-order multidimensional parabolic equations are obtaine

Nonautonomous parabolic equations involving measures

In the first part of this paper, we study abstract parabolic evolution equations involving Banach space-valued measures. These results are applied in the second part to second-order parabolic systems under minimal regularity hypotheses on the coefficients