Weighted Estimates of the Cayley Transform Method for Abstract Differential Equations

Abstract

Abstract We represent the solution u ⁢ ( t ) {u(t)} of an initial value problem (IVP) for the first-order differential equation with an operator coefficient as a series using the Cayley transform of the corresponding operator coefficient and the Laguerre polynomials. In the case of a boundary value problem (BVP) for the second-order differential equation with an operator coefficient, we represent its solution using the Cayley transform and the Meixner-type polynomials. The approximate solution is the truncated sum of N (the discretization parameter) summands. We give the error estimate of these approximations depending on N and the distance of t to the initial point of the time interval or of the spatial argument x to the boundary of the spatial domain.