New parallel algorithms for solving initial and boundary value problems for linear ODEs and their systems on large parallel MIMD computers are proposed. The proposed algorithms are based on dividing a problem in similar so-called local problems, which can be solved independently and in parallel using any known (sequential or parallel) method. The solution is then built as a linear combination of the local solutions. The recurrence relationships (for the case of non-homogeneo us equations) and explicit expressions (for the case of homogeneous equations) for the coefficients of that linear com­ bination are obtained. Three elementary examples, illustrating the idea of the proposed approach, are given. The majority of parallel algorithms were developed for solving algebraic problems and boundary value problems for partial differential equations (PDEs). With the exception of the parallelization of methods of the Runge-Kutta type and their modi­ fications, almost no attention was paid to the development of parallel algorithms for ordinary differential equations (ODEs), and the available literature reflects this state [l]-[5]. However, not every parallel algorithm for solving PDEs is applicable for solving ODEs. Some new parallel algorithms for solving initial and boundary value problems for linear ODEs and their systems are described and illustrated in this paper. The proposed approach is based on two main ideas. 1. The first idea is that, in fact, we always deal with finite intervals when we look for the numerical solution of any initial value problem. Even when the given interval (in the formulation of a problem) is infinite, we can obtain the numerical solution of a problem only for the finite subinterval of the given original infinite interval. So it seems to be natural to apply numerical methods directly to the finite (sub)interval of the researcher's interest.
['Igor Podlubny']
PARALLEL ALGORITHMS FOR INITIAL AND BOUNDARY VALUE PROBLEMS FOR LINEAR ORDINARY DIFFERENTIAL EQUATIONS AND THEIR SYSTEMS