Probabilistic evolution approach for the solution of explicit autonomous ordinary differential equations. Part 1: Arbitrariness and equipartition theorem in Kronecker power series

Gözükırmızı, Coşar; Demіralp, Metin

doi:10.1007/s10910-013-0298-5

Cited by 17 publications

(4 citation statements)

References 19 publications

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“…Therefore, we look at the elements within the term lists. Once we know which node to extend, we use (11) to introduce the new equation. Therefore, Node (3) is created.…”

Section: Classical Quartic Anharmonic Oscillator As An Illustrative Ementioning

confidence: 99%

“…The heuristic distance to the goal is one because the only pair appearing on the right-hand side, but not on the left-hand side, is 2,0. Using (11) on Node (3), it is possible to see that the resulting set can be written in two ways: Node (4) and Node (5). Observing Node (4), its heuristic distance is zero.…”

Section: Classical Quartic Anharmonic Oscillator As An Illustrative Ementioning

confidence: 99%

“…F is not unique due to the linear dependence of the elements of the Kronecker squared vector. Uniqueness may be attained by dividing the contributions equally between the elements corresponding to the linearly-dependent elements of the Kronecker squared vector [11]. The vector of unknowns y(t) may be written as:…”

Section: Brief Recall Of Probabilistic Evolution Theorymentioning

confidence: 99%

“…We will try to avoid any repetition; therefore, we focus on the works that are directly related to this paper. There are several introductory articles on the subject [11,12]. Constancy adding space extension takes second degree multinomial right-hand side functions to purely second degree multinomial right-hand side functions [13].…”

Section: Introductionmentioning

confidence: 99%

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Solving ODEs by Obtaining Purely Second Degree Multinomials via Branch and Bound with Admissible Heuristic

Gözükırmızı

Demіralp

2019

Mathematics

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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.

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“…Therefore, we look at the elements within the term lists. Once we know which node to extend, we use (11) to introduce the new equation. Therefore, Node (3) is created.…”

Section: Classical Quartic Anharmonic Oscillator As An Illustrative Ementioning

confidence: 99%

Section: Classical Quartic Anharmonic Oscillator As An Illustrative Ementioning

confidence: 99%

Section: Brief Recall Of Probabilistic Evolution Theorymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Solving ODEs by Obtaining Purely Second Degree Multinomials via Branch and Bound with Admissible Heuristic

Gözükırmızı

Demіralp

2019

Mathematics

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Probabilistic evolution approach for the solution of explicit autonomous ordinary differential equations. Part 2: Kernel separability, space extension, and, series solution via telescopic matrices

Gözükırmızı

Demіralp

2013

J Math Chem

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Probabilistic evolution approach to the expectation value dynamics of quantum mechanical operators, part I: integral representation of Kronecker power series and multivariate Hausdorff moment problems

Ayvaz

Demіralp

2014

J Math Chem

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Probabilistic evolution approach for the solution of explicit autonomous ordinary differential equations. Part 1: Arbitrariness and equipartition theorem in Kronecker power series

Cited by 17 publications

References 19 publications

Solving ODEs by Obtaining Purely Second Degree Multinomials via Branch and Bound with Admissible Heuristic

Solving ODEs by Obtaining Purely Second Degree Multinomials via Branch and Bound with Admissible Heuristic

Probabilistic evolution approach for the solution of explicit autonomous ordinary differential equations. Part 2: Kernel separability, space extension, and, series solution via telescopic matrices

Probabilistic evolution approach to the expectation value dynamics of quantum mechanical operators, part I: integral representation of Kronecker power series and multivariate Hausdorff moment problems

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