New Difference Schemes for Partial Differential Equations

Abstract

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On source identification problem for a delay parabolic equation

“…The stability estimates in Hölder norms for the solution of this problem are established. Moreover, applying the result of the monograph [43], the high order of accuracy single-step difference schemes for the numerical solution of the source identification problem (1) for a delay parabolic equation with nonlocal conditions can be presented. Of course, the stability estimates for the solution of these difference schemes have been established without any assumptions about the grid steps.…”

“…In the case where the delay term is an operator of the same order with respect to other operator term is studied mainly if H is a Hilbert space (see, for example, [39] and the references given therein). In fact, there are very few papers which allow E to be a general Banach space (see [40][41][42][43][44]) and in these works, authors look only for partial differential equations under regular data. Moreover, approximate solutions of the delay parabolic equations in the case where the delay term is a simple operator of the same order with respect to other operator term were studied recently in papers [45][46][47][48][49].…”

On the problem of determining the parameter of an elliptic equation in a Banach space

“…The proof of Theorem 4 is based on the abstract Theorems 1, 2 and the positivity of the operator A x in C µ (R n ), the structure of the fractional spaces E α ((A x ) 1/2 , C(R n )) [29] and the coercivity inequality for an elliptic operator A x in C µ (R n ) [30]. Second, let Ω = (0, 1) × · · · × (0, 1) be the open cube in the n-dimensional Euclidean space with boundary S, Ω = Ω ∪ S. In [0, T ] × Ω, we consider the mixed boundary value problem for multidimensional elliptic equation…”

“…From positivity of operator A in Banach space E it follows that B = A 1/2 is strongly positive operator in E. Hence, the operator −B is a generator of an analytic semigroup exp{−tB} (t 0) with exponentially decreasing norm (see [29]) as t → ∞, i.e., for some M (B) ∈ [1, +∞), α(B) ∈ (0, +∞) and t > 0, the following estimates are valid:…”

A note on the Taylor’s decomposition on four points for a third-order differential equation

“…The use of Taylor's decomposition on four points permits to construct the difference schemes of the fourth order of accuracy for the approximate solutions of problems (5.4)-(5.6). Operator method of [4] permits to establish the stability of these difference schemes.…”