Publication Info
- Title:
- Exponentially convergent numerical method for the abstract parabolic equation with integral nonlinear nonlocal condition
- Type:
- Article
- Status:
- Submitted
- Journal:
- Ukrainian Mathematical Journal
Abstract
Problem for the first order differential equation with an unbounded operator coefficient in Banach space and integral nonlinear nonlocal condition is considered. An exponentially convergent method is proposed and justified for the numerical solution of this problem under assumption that the mentioned operator coefficient $A$ is strongly positive and some existence and uniqueness conditions are fulfilled. The method is based on the reduction of the given problem to an abstract Hammerstein equation. The later one is discretized by collocation and then solved via the fixed—point iteration method. Each iteration of the method involves Sinc-based numerical evaluation of operator exponential represented by a Dunford-Cauchy integral along hyperbola enveloping the spectrum of $A$. The integral part of nonlocal condition is approximated using the Clenshaw-Curtis quadrature formula.
Codes
All the codes for this article will be made available after the article is approved for publication.