Stability of delay parabolic difference equations

Abstract

In the present paper, the stability of difference schemes for the approximate solution of the initial value problem for delay differential equations with unbounded operators acting on delay terms in an arbitrary Banach space is studied. Theorems on stability of these difference schemes in fractional spaces are established. In practice, the stability estimates in Hölder norms for the solutions of difference schemes for the approximate solutions of the mixed problems for delay parabolic equations are obtained.

Bounded Solutions of Semilinear Time Delay Hyperbolic Differential and Difference Equations

“…are true for any n, n ≥ 1 by mathematical induction. From that and formulas (14) and (15) it follows that…”

“…Ashyralyev and Agirseven [12][13][14][15][16][17][18] studied some initial-boundary value problems for linear delay parabolic differential equations. Theorems on stability and convergence of difference schemes for the numerical solution of initial and boundary value problems for linear parabolic equations with time delay were proved.…”

A tribute to the memory of Pavel Evseevich Sobolevskii (1930-2018)

“…Moreover, this Sobolevskii's operator approach was very important in the study of the solution of the various problems for delay partial differential equations. Actually, Sobolevskii [37] successfully applied theory of interpolation of linear positive operators for delay differential equations of parabolic type. Stability estimates for the solutions of the first and second order of accuracy difference schemes for the approximate solution of this initial value problem for delay differential equations of parabolic type were presented.…”

Stability of the Time-Dependent Identification Problem for the Delay Hyperbolic Equation

“…There is permanently a major interest for the theory of source identification problems for partial differential equations since they have widespread applications in modern physics and technology. For this effort, the stability of various source identification problems for partial differential and difference equations has also been studied extensively by many researchers (see, e.g., [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25] and the references given therein).…”