Sergey Piskarev

A general approximation scheme for attractors of abstract parabolic problems

We consider semilinear problems of the form u′ = Au + f(u), where A generates an exponentially decaying compact analytic C 0-semigroup in a Banach space E, f:E α → E is differentiable globally Lipschitz and bounded (E α = D((−A)α) with the graph norm). Under a very general approximation scheme, we prove that attractors for such problems behave upper semicontinuously. If all equilibrium points are hyperbolic, then there is an odd number of them. If, in addition, all global solutions converge as t → ±∞, then the attractors behave lower semicontinuously. This general approximation scheme includes finite element method, projection and finite difference methods. The main assumption on the approximation is the compact convergence of resolvents, which may be applied to many other problems not related to discretization.

On Well-Posedness of Difference Schemes for Abstract Elliptic Problems in Lp([0, T];E) Spaces

This paper is devoted to the numerical analysis of abstract elliptic differential equations in L p ([0, T];E) spaces. The presentation uses general approximation scheme and is based on C 0-semigroup theory and a functional analysis approach. For the solutions of difference scheme of the second-order accuracy, the almost coercive inequality in spaces with the factor is obtained. In the case of UMD space E n , we establish a coercive inequality for the same scheme in under the condition of R-boundedness.

On a class of time-fractional differential equations

AbstractIn this paper we investigate Cauchy problem for a class of time-fractional differential equation (0.1)

Approximation of Semilinear Fractional Cauchy Problem

\begin{gathered} D_t^\alpha u(t) + c_1 D_t^{\beta _1 } u(t) + \cdots + c_d D_t^{\beta _d } u(t) = Au(t), t > 0, \hfill \\ u^{(j)} (0) = x_j , j = 0, \cdots ,m - 1, \hfill \\ \end{gathered}

Stability of difference schemes for fractional equations

where A is a closed densely defined linear operator in a Banach space X, α > β1 > ... > βd > 0, cj are constants and m = ⌈α⌊. A new type of resolvent family corresponding to well-posedness of (0.1) is introduced. We derive the generation theorems, algebraic equations and approximation theorems for such resolvent families. Moreover, we give the exact solution for a kind of generalized fractional telegraph equations. Some examples are given as illustrations.

On the approximation of fractional resolution families

This paper studies the Crank–Nicolson discretization scheme for abstract differential equations on a general Banach space. We show that a time-varying discretization of a bounded analytic C0-semigroup leads to a bounded discrete-time system. On Hilbert spaces, this result can be extended to all bounded C0-semigroups for which the inverse generator generates a bounded C0-semigroup. The presentation is based on C0-semigroup theory and uses a functional analysis approach.

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

Abstract The semidiscretization methods for solving the Cauchy problem (𝐃 t α u)(t)=Au(t)+J 1-α ft , u ( t ),t∈[0,T],0<α<1,u(0)=u 0 ,

Approximations of Parabolic Equations at the Vicinity of Hyperbolic Equilibrium Point

(\mathbf {D}_{t}^{\alpha }u)(t) = A u(t) + J^{1-\alpha } f\big (t,u(t)\big ), \quad t \in [0,T], 0 < \alpha <1,\qquad u(0) = u^0,