Two-points problem for an evolutional first order equation in Banach space

Two-point nonlocal problem for the first order differential evolution equation with an operator co- efficient 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 unique- ness 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

Approximate solution to abstract differential equations with variable domain

A new exponentially convergent algorithm is proposed for an abstract the first order differential equation with unbounded operator coefficient possessing a variable domain. The algorithm is based on a generalization of the Duhamel integral for vector-valued functions. This technique translates the initial problem to a system of integral equations. Then the system is approximated with exponential accuracy. The theoretical results are illustrated by examples associated with the heat transfer boundary value problems

Approximate solution to abstract differential equations with variable domain

A new exponentially convergent algorithm is proposed for an abstract the first order differential equation with unbounded operator coefficient possessing a variable domain. The algorithm is based on a generalization of the Duhamel integral for vector-valued functions. This technique translates the initial problem to a system of integral equations. Then the system is approximated with exponential accuracy. The theoretical results are illustrated by examples associated with the heat transfer boundary value problems