A probabilistic foundation for dynamical systems: theoretical background and mathematical formulation

Self Cite

This work aims at the solution of two-body problem with Lennard-Jones potential via recently developed Probabilistic Evolution Approach (PEA) proposed by M. Demiralp for explicit ODEs, as final target. PEA has been quite well developed and it is well-known that the conicality in descriptive functions facilitates the construction of the truncation approximants. Conicality, if does not exist, can be obtained by using the space extension method at the expense of an increase in the number of the unknowns and may bring the block triangularity which is another important facility to investigate and control the properties of PEA. In space extension we construct the Hamilton equations of the motion for the system first and then define new appropriate unknown functions depending on the existing unknowns such that the resulting ODEs to be used in PEA become conical. This attempt provides us with the block triangularity at the same time.

Section: Introductionmentioning

confidence: 99%

Probabilistic evolutions in classical dynamics: Conicalization and block triangularization of Lennard-Jones systems

Tunga¹,

Demіralp²

2012

Self Cite

“…In contrast to two body problem these problems cannot be solved analytically in concise manner but must be solved numerically. In this work we will consider a new numerical solution technique to three body problem by using the probabilistic evolution method recently proposed by M. Demiralp [1][2][3][4][5][6][7] and prepare the equations for the application of this approach even though the solution stage will never be explicitly given here since our purpose is solely bringing multinomiality. To this end, the Hamiltonian equations of the system will be defined and the differential equations will be built.…”

Section: Introductionmentioning

confidence: 99%

Multinomialization of the classical three body problem before applying probabilistic evolution approach

Gürvit¹,

Baykara²

2012

This work aims to reshape the equation of motion for a celestial mechanical three body problem for an efficient application of the recently developed Probabilistic Evolution Approach (PEA) for explicit ODEs. It is well known that the multinomiality in descriptive functions facilitates the construction of the truncation approximants. In case of nonexistence, the multinomiality can be obtained by using the space extension method at the expense of an increase in the number of the unknowns and may bring block triangularity which is another important facility to investigate and control the properties of PEA. Before proceeding with space extension we construct the Hamilton equations of motion for the system first and then define new appropriate unknown functions depending on the existing unknowns such that the resulting ODEs to be used in PEA become multinomials. This attempt provides us with block triangularity at the same time.

“…Quite recently we have focused on various issues related to the Probabilistic Evolution Approach (PEA) [3][4][5][6][7]. Amongst these, we have also focused on the Probabilistic Evolution of Van der Pol Equation under certain given initial conditions [5,6].…”

Section: Introductionmentioning

confidence: 99%

“…There we have considered the analyticity of the descriptive functions and used certain power functions as the basis functions. Certain details of the Van der Pol Equations and Ordinary Differential Equation structures are mentioned in [1][2][3][4][5][6]. While formulating the Probabilistic Evolution Approach for Ordinary Differential Equations originally, we were using Taylor expansion, especially in our previous applications.…”

Section: Introductionmentioning

confidence: 99%

Unidirectional semi-analytic continuation in the probabilistic evolution approach to Van der Pol equation

Hunutlu¹,

Baykara²

2012

Probabilistic Evolution Approach (PEA) is a method recently developed for the solution of explicit ODEs. There is now a comprehensive knowledge accumulation for PEA. It is well known that the PEA truncation approximants converge to a unique limit representing the solution of the considered ODE in a finite region of the complex planes of the unknowns. The complement of the union of these regions to entire complex plane(s) is the subject of analytic continuation. In this paper we focus on this issue for the Van der Pol Equation. Paper gives the importance rather to the formalism and conceptuality. True implementations are left to future works.