Pavel E. Sobolevskii

A note on the difference schemes for hyperbolic equations

The initial value problem for hyperbolic equations d 2 u ( t ) / d t 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.

Two new approaches for construction of the high order of accuracy difference schemes for hyperbolic differential equations

We consider the abstract Cauchy problem for differential equation of the hyperbolic type v″(t)

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

Well-posedness of difference elliptic equation

The exact with respect to step h∈(0,1] coercive inequality for solutions in Ch of difference elliptic equation is established.

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

The nonlocal boundary value problem, v ′ ( t ) + A v ( 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 Holder norms for the solution of this problem are proved. The exact Schauders estimates in Holder norms of solutions of the boundary value problem on the range { 0 ≤ t ≤ 1 , x ℝ n } for 2 m -order multidimensional parabolic equations are obtaine.

A note on the difference schemes for hyperbolic-elliptic equations

The nonlocal boundary value problem for hyperbolic-elliptic equation d 2 u ( t ) / d t 2 + A u ( t ) = f ( t ) , ( 0 ≤ t ≤ 1 ) , − d 2 u ( t ) / d t 2 + A u ( 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 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 Holder norms for the solutions of the initial-value problem of the delay differential and difference equations of the parabolic type.

Hardy's inequality for the Stokes problem

Abstract Sharp constant in generalized Hardys inequality for the Stokes problem is obtained in the case of arbitrary convex domain in Rn. It coincides with the same constant in the classical one-dimensional case.

Difference Schemes for Parabolic Equations

Let us consider a differential operator with constant coefficients of the form