AbstractIn the present paper we propose and analyze a class of tensor approaches
for the efficient
numerical solution of a first order differential equation {\psi^{\prime}(t)+A\psi=f(t)}
with an unbounded operator coefficient A.
These techniques are based on a Laguerre polynomial expansions with coefficients which are
powers of the Cayley transform of the operator A.
The Cayley transform under consideration is a useful tool to arrive at the following aims:
(1) to separate time and spatial variables, (2) to switch from the
continuous “time variable” to “the discrete
time variable” and from the study of functions of an unbounded operator to the ones of
a bounded operator, (3) to obtain exponentially accurate approximations.
In the earlier papers of the authors some approximations on the basis
of the Cayley transform and the N-term Laguerre expansions of the accuracy
order {\mathcal{O}(e^{-N})} were proposed and
justified provided that the initial value is analytical for A.
In the present paper we combine the Cayley transform
and the Chebyshev–Gauss–Lobatto interpolation and arrive at an approximation of the accuracy order
{\mathcal{O}(e^{-N})} without restrictions on the input data. The use of the Laguerre expansion or
the Chebyshev–Gauss–Lobatto interpolation allows to separate the time and space variables. The separation of
the multidimensional spatial variable can be achieved by the use of low-rank approximation to
the Cayley transform of the Laplace-like operator that is spectrally close to A.
As a result a quasi-optimal numerical algorithm can be designed.
Laguerre Expansions on Conic Domains