Second Order Equations in Functional Spaces: Qualitative and Discrete Well-Posedness

Ashyralyev, Allaberen; Pastor, Javier; Piskarev, Sergey; Yurtsever, H. A.

doi:10.1155/2015/948321

Cited by 7 publications

(3 citation statements)

References 88 publications

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“…В [57] рассматривалась слабая коэрцитивность для гиперболических дифференциальных уравнений второго порядка в пространствах [59] (см. подробности в [26,58,262]).…”

Section: коэрцитивность в пространствах C αunclassified

Аттракторы, Затенение И Аппроксимация Абстрактных Полулинейных Дифференциальных Уравнений

Piskarev¹,

Ovchinnikov²

2021

Итоги науки и техники. Серия «Современная математика и ее прило

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В обзоре обсуждаются такие разделы теории аппроксимации абстрактных дифференциальных уравнений как аппроксимация аттракторов в случае гиперболических стационарных точек, затенение и аппроксимация дробных по времени полулинейных задач.

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Section: коэрцитивность в пространствах C αunclassified

Аттракторы, Затенение И Аппроксимация Абстрактных Полулинейных Дифференциальных Уравнений

Piskarev¹,

Ovchinnikov²

2021

Итоги науки и техники. Серия «Современная математика и ее прило

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“…The Dirichlet problem for an abstract second order equation (homogeneous or non-homogeneous) has been studied in various papers. Usually it is supposed that −A is a positive operator, we refer to Section 4 of the review paper [2], and the articles quoted therein.…”

Section: Introductionmentioning

confidence: 99%

Dirichlet problem for second-order abstract differential equations

Dore

2020

ejde

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We study the well-posedness in the space of continuous functions of the Dirichlet boundary value problem for a homogeneous linear second-order differential equation u''+Au = 0, where A is a linear closed densely defined operator in a Banach space. We give necessary conditions for the well-posedness, in terms of the resolvent operator of A. In particular we obtain an estimate on the norm of the resolvent at the points k^2, where k is a positive integer, and we show that this estimate is the best possible one, but it is not sufficient for the well-posedness of the problem. Moreover we characterize the bounded operators for which the problem is well-posed. For more information see https://ejde.math.txstate.edu/Volumes/2020/107/abstr.html

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“…The survey paper [44] contains the recent results on the local and nonlocal well-posed problems for second order differential and difference equations. Stability of differential problems for hyperbolic equations and of difference schemes for approximate solution of these problems were presented.…”

Section: Introductionmentioning

confidence: 99%

Bounded Solutions of Semilinear Time Delay Hyperbolic Differential and Difference Equations

Ashyralyev

Agirseven

2019

Mathematics

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In this paper, we study the initial value problem for a semilinear delay hyperbolic equation in Hilbert spaces with a self-adjoint positive definite operator. The mean theorem on the existence and uniqueness of a bounded solution of this differential problem for a semilinear hyperbolic equation with unbounded time delay term is established. In applications, the existence and uniqueness of bounded solutions of four problems for semilinear hyperbolic equations with time delay in unbounded term are obtained. For the approximate solution of this abstract differential problem, the two-step difference scheme of a first order of accuracy is presented. The mean theorem on the existence and uniqueness of a uniformly bounded solution of this difference scheme with respect to time stepsize is established. In applications, the existence and uniqueness of a uniformly bounded solutions with respect to time and space stepsizes of difference schemes for four semilinear partial differential equations with time delay in unbounded term are obtained. In general, it is not possible to get the exact solution of semilinear hyperbolic equations with unbounded time delay term. Therefore, numerical results for the solution of difference schemes for one and two dimensional semilinear hyperbolic equation with time delay are presented. Finally, some numerical examples are given to confirm the theoretical analysis.

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Second Order Equations in Functional Spaces: Qualitative and Discrete Well-Posedness

Cited by 7 publications

References 88 publications

Аттракторы, Затенение И Аппроксимация Абстрактных Полулинейных Дифференциальных Уравнений

Аттракторы, Затенение И Аппроксимация Абстрактных Полулинейных Дифференциальных Уравнений

Dirichlet problem for second-order abstract differential equations

Bounded Solutions of Semilinear Time Delay Hyperbolic Differential and Difference Equations

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