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    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 Ebru
    Probabilistic 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.
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    Probabilistic evolution theory for the solution of explicit autonomous ordinary differential equations: squarified telescope matrices
    (Springer, 2017) Gozukirmizi, Cosar; Kirkin, Melike Ebru; Demiralp, Metin
    Probabilistic 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.