Affiliations: Faculty of Exact and Natural Sciences, I. Javakhishvili Tbilisi State University, Tbilisi, Georgia | School of Mathematics and Information Technologies, University of Georgia, Tbilisi, Georgia | Laboratoire de Modélisation et Simulation Numérique École Supérieure d'Ingénieurs en Électrotechnique et Électronique, Cité Descartes, Noisy-le-Grand Cedex, France
Note: [] Corresponding author: M. Avalishvili, School of Mathematics and Information Technologies, University of Georgia, 77a Kostava str., Tbilisi 0175, Georgia. E-mail: [email protected].
Abstract: In this paper we consider the nonclassical initial and initial-boundary value problems for second-order evolution equations with nonlocal initial conditions. We study the existence and uniqueness, in suitable abstract spaces, of solution of vector-valued distributions for such problems. We also present an algorithm of approximation of the solution of nonclassical problem with nonlocal initial conditions by a sequence of solutions of the corresponding classical problems. As an example we investigate the case of nonclassical hyperbolic equations and systems. In this particular case we show that the existence and uniqueness of solutions rely on algebraic properties of the ratios of values of the time variable in the nonlocal initial conditions and sizes of the spatial domain.
Keywords: abstract second-order evolution equations, initial-boundary value problems for hyperbolic equations and systems, nonlocal initial conditions
