Two-points problem for an evolutional first order equation in Banach space
T. Ju. Bohonova, V. B. Vasylyk
TL;DR
The paper addresses two-point nonlocal problems for first-order evolution equations with unbounded, time-dependent operator coefficients in a Banach space, proposing an exponentially convergent numerical scheme. By exploiting strong positivity of the operator and evolution operators $U(t,s)$, it derives existence/uniqueness criteria for small $|\alpha|$ and provides an explicit solution representation in terms of $U(t,s)$. The numerical method uses a Chebyshev–Gauss–Lobatto discretization after a time transformation, yielding a linear system that can be solved by a contraction iteration with provable stability and error bounds. Numerical experiments on a heat-type PDE with manufactured solutions verify exponential convergence and demonstrate practical efficiency for nonlocal evolution problems.
Abstract
Two-points nonlocal problem for the first order differential evolution equation with an operator coefficient in a Banach space $X$ is considered. An exponentially convergent algorithm is proposed and justified in assumption that the operator coefficient is strongly positive and some existence and uniqueness conditions are fulfilled. This algorithm leads to a system of linear equations that can be solved by fixed-point iteration. The algorithm provides exponentially convergence in time that in combination with fast algorithms on spatial variables can be efficient treating such problems. The efficiency of the proposed algorithms is demonstrated by numerical examples.