On stability and convergence of a three-layer semi-discrete scheme for an abstract analogue of the Ball integro-differential equation

We consider the Cauchy problem for a second-order nonlinear evolution equation in a Hilbert space. This equation represents the abstract generalization of the Ball integro-differential equation. The general nonlinear case with respect to terms of the equation which include a square of a norm of a gr...

Celý popis

Uloženo v:
Podrobná bibliografie
Vydáno v:Journal of mathematical analysis and applications Ročník 518; číslo 1; s. 126664
Hlavní autoři: Rogava, Jemal, Tsiklauri, Mikheil, Vashakidze, Zurab
Médium: Journal Article
Jazyk:angličtina
Vydáno: Elsevier Inc 01.02.2023
Témata:
Abstract analogue of beam equation
Cauchy problem
Chebyshev polynomials
Nonlinear integro-differential equation
Stability and convergence
Three–layer semi–discrete scheme
Nonlinear integro-differential equation
Abstract analogue of beam equation
Three–layer semi–discrete scheme
Stability and convergence
Chebyshev polynomials
Cauchy problem
ISSN:0022-247X, 1096-0813
On-line přístup:Získat plný text
Tagy: Přidat tag
Žádné tagy, Buďte první, kdo vytvoří štítek k tomuto záznamu!
Popis
Shrnutí:We consider the Cauchy problem for a second-order nonlinear evolution equation in a Hilbert space. This equation represents the abstract generalization of the Ball integro-differential equation. The general nonlinear case with respect to terms of the equation which include a square of a norm of a gradient is considered. A three-layer semi-discrete scheme is proposed in order to find an approximate solution. In this scheme, the approximation of nonlinear terms that are dependent on the gradient is carried out by using an integral mean. We show that the solution of the nonlinear discrete problem and its corresponding difference analogue of a first-order derivative is uniformly bounded. For the solution of the corresponding linear discrete problem, it is obtained high-order a priori estimates by using two-variable Chebyshev polynomials. Based on these estimates we prove the stability of the nonlinear discrete problem. For smooth solutions, we provide error estimates for the approximate solution. An iteration method is applied in order to find an approximate solution for each temporal step. The convergence of the iteration process is proved.
ISSN:0022-247X
1096-0813
DOI:10.1016/j.jmaa.2022.126664