Allaberen Ashyralyev

New difference schemes for partial differential equations

1 Linear Difference Equations.- 1.1 Difference Equations of the First Order.- 1.2 Difference Equations of the Second Order.- 1.3 Difference Equations with Constant Coefficients.- 2 Difference Schemes for First-Order Differential Equations.- 2.1 Single-Step Exact Difference Scheme and Its Applications.- 2.2 Taylors Decomposition on Two Points and Its Applications.- 3 Difference Schemes for Second-Order Differential Equations.- 3.1 Two-Step Exact Difference Scheme and Its Applications.- 3.2 Taylors Decomposition on Three Points and Its Applications.- 4 Partial Differential Equations of Parabolic Type.- 4.1 A Cauchy Problem. Well-posedness.- 4.2 Difference Schemes Generated by an Exact Difference Scheme.- 4.3 Single-Step Difference Schemes Generated by Taylors Decomposition.- 5 Partial Differential Equations of Elliptic Type.- 5.1 A Boundary-Value Problem. Well-posedness.- 5.2 Difference Schemes Generated by an Exact Difference Scheme.- 5.3 Two-Step Difference Schemes Generated by Taylors Decomposition.- 6 Partial Differential Equations of Hyperbolic Type.- 6.1 A Cauchy Problem.- 6.2 Difference Schemes Generated by an Exact Difference Scheme.- 6.3 Two-Step Difference Schemes Generated by Taylors Decomposition.- 7 Uniform Difference Schemes for Perturbation Problems.- 7.1 A Cauchy Problem for Parabolic Equations.- 7.2 A Boundary-Value Problem for Elliptic Equations.- 7.3 A Cauchy Problem for Hyperbolic Equations.- 8 Appendix: Delay Parabolic Differential Equations.- 8.1 The Initial-Value Differential Problem.- 8.2 The Difference Schemes.- Comments on the Literature.

Well-posedness of parabolic difference equations

This monograph is devoted to the construction of highly accurate difference schemes for parabolic boundary value problems, based on Pade approximations. The investigation is based on a new notion of positivity of difference operators in Banach spaces, which allows one to deal with difference schemes of arbitrary order of accuracy. Establishing coercivity inequalities allows one to obtain sharp - that is, two-sided - estimates of convergence rates. The proofs are based on results in interpolation theory of linear operators. The book should be of value to professional mathematicians, as well as advanced students in the fields of functional analysis and partial differential equations.

On Well-Posedness of the Nonlocal Boundary Value Problems for Elliptic Equations

Abstract The nonlocal boundary value problem in an arbitrary Banach space E with the positive operator A is considered. The well-posedness of this boundary value problem in the spaces of smooth functions is established. The new exact Schauders estimates of solutions of the boundary value problems for elliptic equations are obtained.

A Note on the Difference Schemes of the Nonlocal Boundary Value Problems for Hyperbolic Equations

Abstract The nonlocal boundary-value problem for hyperbolic equations in a Hilbert space H with the self-adjoint positive definite operator A is considered. Applying the operator approach, we establish the stability estimates for solution of this nonlocal boundary-value problem. In applications, the stability estimates for the solution of the nonlocal boundary value problems for hyperbolic equations are obtained. The first and second order of accuracy difference schemes generated by the integer power of A for approximately solving this abstract nonlocal boundary-value problem are presented. The stability estimates for the solution of these difference schemes are obtained. The theoretical statements for the solution of this difference schemes are supported by the results of numerical experiments.

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

Fractional spaces generated by the positive differential and difference operators in a Banach space

The structure of the fractional spaces E α,q,(L q[0, 1], A x) generated by the positive differential operator A x defined by the formula A x u = −a(x) d 2 u/dx 2 + δu, with domain D(A x) = {u ∈ C (2)[0, 1] : u(0) = u(1), u′(0) = u′(1)} is investigated. It is established that for any 0 < α < 1/2 the norms in the spaces E α,q(L q[0, 1],A x) and W q 2α [0, 1] are equivalent. The positivity of the differential operator A x in W q 2α [0, 1](0 ≤ α < 1/2) is established. The discrete analogy of these results for the positive difference operator A h x a second order of approximation of the differential operator A x, defined by the formula

ON WELL-POSEDNESS OF DIFFERENCE SCHEMES FOR ABSTRACT PARABOLIC EQUATIONS IN L P ([0,T]; E) SPACES

A_h^x u^h = \left\{ { - a\left( {x_k } \right)\frac{{u_{k + 1} - 2u_k + u_{k - 1} }} {{h^2 }} + \delta u_k } \right\}_1^{M - 1} ,u_h = \left\{ {u_k } \right\}_0^M ,Mh = 1