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Öğe Classical Symmetric Fourth Degree Potential Systems in Probabilistic Evolution Theoretical Perspective: Most Facilitative Conicalization and Squarification of Telescope Matrices(Amer Inst Physics, 2017) Gozukirmizi, Cosar; Kirkin, Melike EbruProbabilistic evolution theory (PREVTH) provides a powerful framework for the solution of initial value problems of explicit ordinary differential equation sets with second degree multinomial right hand side functions. The use of the recursion between squarified telescope matrices provides the opportunity to obtain accurate results without much effort. Convergence may be considered as one of the drawbacks of PREVTH. It is related to many factors: the initial values and the coefficients in the right hand side functions are the most apparent ones. If a space extension is utilized before PREVTH, the convergence of PREVTH may also be affected by how the space extension is performed. There are works about implementations related to probabilistic evolution and how to improve the convergence by methods like analytic continuation. These works were written before squarification was introduced. Since recursion between squarified telescope matrices has given us the opportunity to obtain results corresponding to relatively higher truncation levels, it is important to obtain and analyze results related to certain problems in different areas of engineering. This manuscript may be considered to be in a series of papers and conference proceedings which serves for this purpose.Öğe Probabilistic Evolution Theory for Explicit Autonomous ODEs: Simplifying the Factorials, Cauchy Product Folding and Kronecker Product Decomposition(Amer Inst Physics, 2018) Gozukirmizi, CosarProbabilistic evolution theory forms a framework for the solution of explicit ODEs. The squarification concept and the recursion between the squarified telescope matrices (or the images of initial vectors under the squarified telescope matrices) make the method efficient in terms of time and space necessities. This work is designed for further improvements. The new concepts in this work are simplifying the factorials, Cauchy product folding, the use of Kronecker product decomposition within Probabilistic evolution theory (PREVTH) and condensation. Simplifying the factorials is about embedding the factorial that is appearing in the series expansion into the recursion so that number of calculations is reduced and numerical stability is improved. Cauchy product folding is about the Cauchy Kronecker product of two vectors. If the vectors change places in Kronecker product, the new result is a permutation of the original result. This property is used so that the number of terms in the series is halved. Also, the possibilities of the use of Kronecker product decomposition on the rectangular matrix are investigated in detail. Lastly, in condensation, the effect of Cauchy product folding which causes equivalent columns in the rectangular matrix is utilized so that smaller matrices and vectors may be used in the recursion.Öğe Probabilistic evolution theory for explicit autonomous ordinary differential equations: recursion of squarified telescope matrices and optimal space extension(Springer, 2018) Gozukirmizi, Cosar; Demiralp, MetinProbabilistic evolution theory facilitates the solution of initial value problem of explicit autonomous ordinary differential equations with second degree multinomial right hand side functions. Its formulation has components we call telescope matrices. The matrices grow in size very rapidly and has many zeroes and repeating structures. In order to avoid the computational complexity coming from telescope matrices, squarified telescope matrices are utilized. Their calculation is through a recursion. This recursion has been used in several works by the authors and their colleagues but its proof was not given. This work gives the proof of the recursion and all the surrounding details. A second purpose of this work is to provide a method for most facilitative (optimal) space extension. Space extension is needed for using probabilistic evolution theory when degree of multinomiality of the right hand side functions is more than two. For this purpose, an approach using method of exhaustion (brute-force) is proposed.Öğe Probabilistic evolution theory for the solution of explicit autonomous ordinary differential equations: squarified telescope matrices(Springer, 2017) Gozukirmizi, Cosar; Kirkin, Melike Ebru; Demiralp, MetinProbabilistic evolution theory (PREVTH) is used for the solution of initial value problems of first order explicit autonomous ordinary differential equation sets with second degree multinomial right hand side functions. It is an approximation method based on Kronecker power series: a rewriting of multivariate Taylor series using matrices having certain flexible parameters. Kronecker power series have matrices which are called telescope matrices: matrices where j is the index of summation. The additive terms of each telescope matrix is formed through Kronecker product from both sides by Kronecker powers of identity matrices. Recently, squarification is proposed in order to avoid the growing of the matrices in size at each additive term of the series. This paper explains the squarification procedure: the procedure used in order to avoid Kronecker multiplications within PREVTH so that the sizes of the matrices do not grow and so that the amount of necessary computation is reduced. The recursion between squarified matrices is also given. As a numerical application, the solution of a H,non-Heiles system is provided.Öğe Squarification of Telescope Matrices in the Probabilistic Evolution Theoretical Approach to the Two Particle Classical Mechanics as an Illustrative Implementation(Amer Inst Physics, 2017) Gozukirmizi, Cosar; Tataroglu, ElifIn this work, the use of probabilistic evolution theory for the solution of two particle classical mechanics problem is under consideration. Using the separation of the mass center and some algebra, it is possible to reduce the problem to the solution of the initial value problem which has two differential equations and two unknown functions. This is the starting point of this manuscript and the steps up to this point is given in a companion paper. Then, space extension, constancy adding space extension, probabilistic evolution theory with squarification are utilized to form a numerical approximation. These steps and the results are detailed. Therefore, a complete application of probabilistic evolution theory is provided and the results are compared to the results obtained by fourth order Runge-Kutta method.
