25th Anniversary of TKSL

In recent years, intensive research has been done at the Brno University of Technology Faculty of Information Technology Department of Intelligent Systems in the field of numerical solutions of systems of ordinary and partial differential equations. The basic numerical method employed is the so-called Modem Taylor Series Method (MTSM). It has been described, studied, and numerous aspects have been investigated such as processing in parallel systems. Also a simulation system TKSL has been developed which is based on the Taylor series method. For some results see [1], [2]. Although there have been considerable practical results, theoretical issues are yet to be investigated since the underlying method has been devised by analogy with analogue solvers of such systems. In this paper we provide only a basic idea of a theoretical background. The MTSM is based on a transformation of the initial problem into another initial problem with polynomials on the right-hand sides. This is a precondition for a Taylor series method to be successfully applied to the task of finding a numerical solution. The solution of the transformed initial problem then includes the solution of the original original system. A transformation similar to that described in this paper can be found in [3]. Practical aspects are discussed in [4], and [1]. An outline of the special type transformations for functions commonly encountered in initial problems can be found in [5]. The MTSM uses various types of recurrent formulae that can be used to calculate the Taylor series terms of the unknown functions. They include the detection of polynomial solutions. Another type of recurrent formulae clears the way for parallel programs to be used for solving. This type might also be of some interest for further research since it defines another type of initial problem where the right-hand sides include only powers of unknown functions.