A new one-step method with three intermediate points in a variable step-size mode for stiff differential systems.

[EN]This work introduces a new one-step method with three intermediate points for solving stiff differential systems. These types of problems appear in different disciplines and, in particular, in problems derived from chemical reactions. In fact, the term “stiff”’ was coined by Curtiss and Hirschfelder in an article on problems of chemical kinetics (Hirschfelder, Proc Natl Acad Sci USA 38:235–243, 1952). The techniques of interpolation and collocation are used in the construction of the scheme. We consider a suitable polynomial to approximate the theoretical solution of the problem under consideration. The basic properties of the new scheme are analyzed. An embedded strategy is adopted to formulate the proposed scheme in a variable stepsize mode to get better performance. Finally, some models of initial-value problems, including ordinary and time-dependent partial differential equations, are solved numerically to assess the performance and efficiency of the proposed technique, with applications to real-world problems

Numerical methods for solving initial value problems on the Infinity Computer

New algorithms for the numerical solution of Ordinary Differential Equations (ODEs) with initial condition are proposed. They are designed for work on a new kind of a supercomputer – the Infinity Computer, – that is able to deal numerically with finite, infinite and infinitesimal numbers. Due to this fact, the Infinity Computer allows one to calculate the exact derivatives of functions using infinitesimal values of the stepsize. As a consequence, the new methods described in this paper are able to work with the exact values of the derivatives, instead of their approximations

Numerical infinities and infinitesimals: Methodology, applications, and repercussions on two Hilbert problems

In this survey, a recent computational methodology paying a special attention to the separation of mathematical objects from numeral systems involved in their representation is described. It has been introduced with the intention to allow one to work with infinities and infinitesimals numerically in a unique computational framework in all the situations requiring these notions. The methodology does not contradict Cantor’s and non-standard analysis views and is based on the Euclid’s Common Notion no. 5 “The whole is greater than the part” applied to all quantities (finite, infinite, and infinitesimal) and to all sets and processes (finite and infinite). The methodology uses a computational device called the Infinity Computer (patented in USA and EU) working numerically (recall that traditional theories work with infinities and infinitesimals only symbolically) with infinite and infinitesimal numbers that can be written in a positional numeral system with an infinite radix. It is argued that numeral systems involved in computations limit our capabilities to compute and lead to ambiguities in theoretical assertions, as well. The introduced methodology gives the possibility to use the same numeral system for measuring infinite sets, working with divergent series, probability, fractals, optimization problems, numerical differentiation, ODEs, etc. (recall that traditionally different numerals lemniscate; Aleph zero, etc. are used in different situations related to infinity). Numerous numerical examples and theoretical illustrations are given. The accuracy of the achieved results is continuously compared with those obtained by traditional tools used to work with infinities and infinitesimals. In particular, it is shown that the new approach allows one to observe mathematical objects involved in the Hypotheses of Continuum and the Riemann zeta function with a higher accuracy than it is done by traditional tools. It is stressed that the hardness of both problems is not related to their nature but is a consequence of the weakness of traditional numeral systems used to study them. It is shown that the introduced methodology and numeral system change our perception of the mathematical objects studied in the two problems