Solving ODEs by Obtaining Purely Second Degree Multinomials via Branch and Bound with Admissible Heuristic†
Computer Engineering Department, Beykent University, Ayazaga, Istanbul 34485, Turkey
Computational Science and Engineering Department, Istanbul Technical University, Ayazaga, Istanbul 34469, Turkey
Author to whom correspondence should be addressed.
Part of this paper is an extended version of the conference proceeding published in “Gözükırmızı, Coşar. Probabilistic evolution theory for explicit autonomous ODEs: Simplifying the factorials, Cauchy product folding and Kronecker product decomposition. AIP Conference Proceedings 2018, 2046, 020034, doi:10.1063/1.5081554”.
Mathematics 2019, 7(4), 367; https://doi.org/10.3390/math7040367
Received: 6 February 2019 / Revised: 12 April 2019 / Accepted: 16 April 2019 / Published: 22 April 2019
(This article belongs to the Section Mathematics and Computer Science)
Probabilistic evolution theory (PREVTH) forms a framework for the solution of explicit ODEs. The purpose of the paper is two-fold: (1) conversion of multinomial right-hand sides of the ODEs to purely second degree multinomial right-hand sides by space extension; (2) decrease the computational burden of probabilistic evolution theory by using the condensed Kronecker product. A first order ODE set with multinomial right-hand side functions may be converted to a first order ODE set with purely second degree multinomial right-hand side functions at the expense of an increase in the number of equations and unknowns. Obtaining purely second degree multinomial right-hand side functions is important because the solution of such equation set may be approximated by probabilistic evolution theory. A recent article by the authors states that the ODE set with the smallest number of unknowns can be found by searching. This paper gives the details of a way to search for the optimal space extension. As for the second purpose of the paper, the computational burden can be reduced by considering the properties of the Kronecker product of vectors and how the Kronecker product appears within the recursion of PREVTH: as a Cauchy product structure.