Conference paperKUNOVSKÝ Jiří, SEHNALOVÁ Pavla and ŠÁTEK Václav. Explicit and Implicit Taylor Series Based Computations. In: 8th International Conference of Numerical Analysis and Applied Mathematics. Tripolis: American Institute of Physics, 2010, pp. 587590. ISBN 9780735408319.  Publication language:  english 

Original title:  Explicit and Implicit Taylor Series Based Computations 

Title (cs):  Počítání pomocí explicitní a implicitní Taylorovy řady 

Pages:  587590 

Proceedings:  8th International Conference of Numerical Analysis and Applied Mathematics 

Conference:  8th International Conference of Numerical Analysis and Applied Mathematics 

Place:  Tripolis, GR 

Year:  2010 

ISBN:  9780735408319 

Publisher:  American Institute of Physics 

Keywords 

Differential equations, Taylor series method, TKSL, Stiff systems 
Annotation 

This paper deals with computer simulations of continuous systems. The research group "High performance computing" has been working on extremely exact and fast solutions of homogenous differential equations, nonlinear ordinary and partial differential equations, stiff systems, large systems of algebraic equations, real time simulations and corresponding software and hardware (parallel) implementations since 1980. The Modern Taylor Series Method (MTSM) developed at our university is an original mathematical method which uses the Taylor series method for solving differential equations in a nontraditional way. Unfortunately, it is easier said than done as there are some peculiar systems of differential equations, which cannot be solved by commonly used (explicit) methods  the stiff systems.While the definition of this kind of systems is intuitively clear to the mathematicians the exact definition has not been specified yet. Often unnoticed, stiff systems showed up too often in practice such as in simulations of electrical circuits, chemical reactions and so on. To solve this kind of problems we can use for example multiple arithmetic or implicit numerical methods. In this paper we compare explicit and implicit Taylor series method in solution of the wellknown Dahlquist's equation. We focus on stability and convergence of corresponding computations. Both explicit and implicit Taylor series methods give very successful results for the Dahlquist's equation and it can be expected that similar results will be obtained for sophisticated problems. 
BibTeX: 

@INPROCEEDINGS{
author = {Ji{\v{r}}{\'{i}} Kunovsk{\'{y}} and Pavla
Sehnalov{\'{a}} and V{\'{a}}clav {\v{S}}{\'{a}}tek},
title = {Explicit and Implicit Taylor Series Based
Computations},
pages = {587590},
booktitle = {8th International Conference of Numerical Analysis and
Applied Mathematics},
year = 2010,
location = {Tripolis, GR},
publisher = {American Institute of Physics},
ISBN = {9780735408319},
language = {english},
url = {http://www.fit.vutbr.cz/research/view_pub.php.en?id=9352}
} 
