Sergey Piskarev

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We present an analysis of the discretization methods for solving in Banach space E the semilinear Cauchy problem u 00 (t) = Au(t) + f(t; u(t)); u(0) = u 0 ; u 0 (0) = u 1 ; with operator A, which generate cosine operator function. We consider the existence of the solution in the case of compact function f and give analysis of semidiscretization on a general(More)
This paper is devoted to the numerical analysis of abstract parabolic problem u′ t Au t ; u 0 u0, with hyperbolic generator A. We are developing a general approach to establish a discrete dichotomy in a very general setting in case of discrete approximation in space and time. It is a well-known fact that the phase space in the neighborhood of the hyperbolic(More)
and Applied Analysis 3 2. Preliminaries Let V andH be complex Hilbert spaces forming Gelfand triple V ⊂ H ⊂ V ∗ with pivot space H. The norms of V , H and V ∗ are denoted by || · ||, | · |, and || · ||∗, respectively. The inner product inH is defined by ·, · . The embeddings V ↪→ H ↪→ V ∗ 2.1 are continuous. Then the following inequality easily follows:(More)
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