On uniform difference schemes of a high order of approximation for evolution equations with a small parameter

We consider initial value problems for evolution equations in Banach space E with a small parameter E multiplying the time-derivative term. For such problems we construct difference schemes of a high order of approximation that are uniform with respect to the parameter in (0,1).     

Stable difference schemes for the hyperbolic problems subject to nonlocal boundary conditions with self-adjoint operator

In the present paper the first and second orders of accuracy difference schemes for the numerical solution of multidimensional hyperbolic equations with nonlocal boundary and Dirichlet conditions are presented. The stability estimates for the solution of difference schemes are obtained. A method is used for solving these difference schemes in the case of one dimensional hyperbolic equation.     

On the numerical solution of hyperbolic PDEs with variable space operator

The first and second order of accuracy in time and second order of accuracy in the space variables difference schemes for the numerical solution of the initial‐boundary value problem for the multidimensional hyperbolic equation with dependent coefficients are considered. Stability estimates for the solution of these difference schemes and for the first and second order difference derivatives are obtained. Numerical methods are proposed for solving the one‐dimensional hyperbolic partial differential equation. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2009     

A note on fractional derivatives and fractional powers of operators

Definitions of fractional derivatives and fractional powers of positive operators are considered. The connection of fractional derivatives with fractional powers of positive operators is presented. The formula for fractional difference derivative is obtained.     

On the problem of determining the parameter of a parabolic equation

We study the boundary-value problem of determining the parameter p of a parabolic equation

The Structure of Fractional Spaces Generated by a Two-dimensional Difference Operator in a Half Plane

Ukrainian Mathematical Journal - We consider a difference-operator approximation $$ {A}_h^x $$ of the differential operator $$...     

On a difference scheme of fourth order of accuracy for the Bitsadze–Samarskii type nonlocal boundary value problem

The Bitsadze–Samarskii type nonlocal boundary value problem for the differential equation in a Hilbert space H with the self‐adjoint positive definite operator A with a closed domain D(A) ? H is considered. Here, f(t) be a given abstract continuous function defined on [0,1] with values in H, φ and ψ be the elements of D(A), and λ_j are the numbers from the set [0,1]. The well‐posedness of the problem in Hölder spaces with a weight is established. The coercivity inequalities for the solution of the nonlocal boundary value problem for elliptic equations are obtained. The fourth order of accuracy difference scheme for approximate solution of the problem is presented. The well‐posedness of this difference scheme in difference analogue of Hölder spaces is established. For applications, the stability, the almost coercivity, and the coercivity estimates for the solutions of difference schemes for elliptic equations are obtained. Mathematical Methods in the Applied Sciences. Copyright © 2012 John Wiley & Sons, Ltd.