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 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.
J. Rogava et al., "On Stability and Convergence of a Three-Layer Semi-Discrete Scheme for an Abstract Analogue of the Ball Integro-Differential Equation," Journal of Mathematical Analysis and Applications, vol. 518, no. 1, article no. 126664, Elsevier, Feb 2023.
The definitive version is available at https://doi.org/10.1016/j.jmaa.2022.126664
Electrical and Computer Engineering
Keywords and Phrases
Abstract Analogue of Beam Equation; Cauchy Problem; Chebyshev Polynomials; Nonlinear Integro-Differential Equation; Stability and Convergence; Three–layer Semi–discrete Scheme
International Standard Serial Number (ISSN)
Article - Journal
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01 Feb 2023