Simulation of the linear mechanical oscillator on GPU

U ovom kratkom radu mi smo opisali neke metode za rešavanje stohastičkih diferencijalnih jednačina koje se javljaju u mehanici. Pre svega nas zanimaju mehaničke oscilacije koje nastaju slučajnom pobudom. Slučajna pobuda u obliku belog šuma se najčešće koristi za teorijska i praktična razmatranja, zato i mi biramo pobudu u obliku belog šuma. Standardni oblik jednačine mehaničkog oscilatora je zapisan u obliku koji je pogodan za numeričko rešavanje. Na kraju dajemo neke elemente pisanja programa na GPU.In this short paper we are describing some methods for solving stochastic differential equations which appears in mechanics. We are concerned with solutions of mechanical oscillators which are subject to random excitation. We are primarly interested in solving stochastic differential equations excited with white noise. We rewrite the random mechanical oscillator equation in the form suitable for applying numerical procedures for solving it. Finally, we give some design notes on the methods used to implement numerical simulation of the graphical processing units

Simulation of the linear mechanical oscillator on GPU

U ovom kratkom radu mi smo opisali neke metode za rešavanje stohastičkih diferencijalnih jednačina koje se javljaju u mehanici. Pre svega nas zanimaju mehaničke oscilacije koje nastaju slučajnom pobudom. Slučajna pobuda u obliku belog šuma se najčešće koristi za teorijska i praktična razmatranja, zato i mi biramo pobudu u obliku belog šuma. Standardni oblik jednačine mehaničkog oscilatora je zapisan u obliku koji je pogodan za numeričko rešavanje. Na kraju dajemo neke elemente pisanja programa na GPU.In this short paper we are describing some methods for solving stochastic differential equations which appears in mechanics. We are concerned with solutions of mechanical oscillators which are subject to random excitation. We are primarly interested in solving stochastic differential equations excited with white noise. We rewrite the random mechanical oscillator equation in the form suitable for applying numerical procedures for solving it. Finally, we give some design notes on the methods used to implement numerical simulation of the graphical processing units

A set of programs for analysis of kinetic and equilibrium data

A program package that can be used for analysis of a wide range of kinetic and equilibrium data is described. The four programs were written in Turbo Pascal and run on PC, XT, AT and compatibles. The first of the programs allows the user to fit data with 16 predefined and one user-defined function, using two different non-linear least-squares procedures. Two additional programs are used to test both the evaluation of model functions and the least-squares fits. One of these programs uses two simple procedures to generate a Gaussian-distributed random variable that is used to simulate the experimental error of measurements. The last program simulates kinetics described by differential equations that cannot be solved analytically, using numerical integration. This program helps the user to judge the validity of steady-state assumptions or treatment of kinetic measurements as relaxation

Statistics of non-linear stochastic dynamical systems under L\'evy noises by a convolution quadrature approach

This paper describes a novel numerical approach to find the statistics of the non-stationary response of scalar non-linear systems excited by L\'evy white noises. The proposed numerical procedure relies on the introduction of an integral transform of Wiener-Hopf type into the equation governing the characteristic function. Once this equation is rewritten as partial integro-differential equation, it is then solved by applying the method of convolution quadrature originally proposed by Lubich, here extended to deal with this particular integral transform. The proposed approach is relevant for two reasons: 1) Statistics of systems with several different drift terms can be handled in an efficient way, independently from the kind of white noise; 2) The particular form of Wiener-Hopf integral transform and its numerical evaluation, both introduced in this study, are generalizations of fractional integro-differential operators of potential type and Gr\"unwald-Letnikov fractional derivatives, respectively.Comment: 20 pages, 5 figure

The Hypotheses on Expansion of Iterated Stratonovich Stochastic Integrals of Arbitrary Multiplicity and Their Partial Proof

In this review article we collected more than ten theorems on expansions of iterated Ito and Stratonovich stochastic integrals, which have been formulated and proved by the author. These theorems open a new direction for study of iterated Ito and Stratonovich stochastic integrals. The expansions based on multiple and iterated Fourier-Legendre series as well as on multiple and iterated trigonomectic Fourier series converging in the mean and pointwise are presented in the article. Some of these theorems are connected with the iterated stochastic integrals of multiplicities 1 to 5. Also we consider two theorems on expansions of iterated Ito stochastic integrals of arbitrary multiplicity $k$ $(k\in\mathbb{N})$ based on generalized multiple Fourier series converging in the sense of norm in Hilbert space $L_2([t, T]^k)$ as well as two theorems on expansions of iterated Stratonovich stochastic integrals of arbitrary multiplicity $k$ $(k\in\mathbb{N})$ based on generalized iterated Fourier series converging pointwise. On the base of the presented theorems we formulate 3 hypotheses on expansions of iterated Stratonovich stochastic integrals of arbitrary multiplicity $k$ $(k\in\mathbb{N})$ based on generalized multiple Fourier series converging in the sense of norm in Hilbert space $L_2([t, T]^k).$ The mentioned iterated Stratonovich stochastic integrals are part of the Taylor-Stratonovich expansion. Moreover, the considered expansions from these 3 hypotheses contain only one operation of the limit transition and substantially simpler than their analogues for iterated Ito stochastic integrals. Therefore, the results of the article can be useful for the numerical integration of Ito stochastic differential equations. Also, the results of the article were reformulated in the form of theorems of the Wong-Zakai type for iterated Stratonovich stochastic integrals.Comment: 35 pages. Section 12 was added. arXiv admin note: text overlap with arXiv:1712.09516, arXiv:1712.08991, arXiv:1802.04844, arXiv:1801.00231, arXiv:1712.09746, arXiv:1801.0078