Stability and Convergence of the Modern Taylor Series Method

Stability, Convergence, Modern Taylor Series Method, Differential equations, Continuous system modelling

The paper deals with extremely exact, stable and fast numerical solutions of systems of differential equations. In a natural way, it also involves solutions of problems that can be transformed to solving a system of differential equations.
The project is based on an original mathematical method which uses the Taylor series method for solving differential equations.
The Taylor Series Method is based on a recurrent calculation of the Taylor series terms for each time interval. Thus the complicated calculation of higher order derivatives (much criticized in the literature) need not be performed but rather the value of each Taylor series term is numerically calculated. Another typical algorithm is the convolution operation. Stability and convergence of the numerical integration methods when the Dahlquist problem is solved, Taylorian initial problems with automatic transformation, stability and convergence of a system of linear algebraic equations and stability and convergence when algebraic and transcendental equations are solved is discussed in this paper.