Modelling a solution of a hyperbolic equation with random initial conditions

Authors:
Yu. V. Kozachenko and G. I. Slivka

Translated by:
Oleg Klesov

Original publication:
Teoriya Imovirnostei ta Matematichna Statistika, tom **74** (2006).

Journal:
Theor. Probability and Math. Statist. **74** (2007), 59-75

MSC (2000):
Primary 60G35; Secondary 35L20

Published electronically:
June 29, 2007

MathSciNet review:
2336779

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A new method is proposed in this paper to construct models for solutions of boundary problems for hyperbolic equations with random initial conditions. We assume that initial conditions are strictly sub-Gaussian random fields (in particular, Gaussian random fields with zero mean). The models approximate solutions with a given accuracy and reliability in the uniform metric.

**1.**V. V. Buldygin and Yu. V. Kozachenko,*Metric characterization of random variables and random processes*, Translations of Mathematical Monographs, vol. 188, American Mathematical Society, Providence, RI, 2000. Translated from the 1998 Russian original by V. Zaiats. MR**1743716****2.**Yu. V. Kozachenko and A. O. Pashko,*Modelling Stochastic Processes*, ``Kyiv University'', Kyiv, 1999. (Ukrainian)**3.**Yu. V. Kozachenko and G. Ī. Slivka,*Justification of the Fourier method for a hyperbolic equation with random initial conditions*, Teor. Ĭmovīr. Mat. Stat.**69**(2003), 63–78 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist.**69**(2004), 67–83 (2005). MR**2110906**, 10.1090/S0094-9000-05-00615-0**4.**N. S. Koshlyakov, M. M. Smirnov, and E. B. Gliner,*Differential equations of mathematical physics*, Translated by Scripta Technica, Inc.; translation editor: Herbert J. Eagle, North-Holland Publishing Co., Amsterdam; Interscience Publishers John Wiley and Sons New York, 1964. MR**0177179****5.**G. N. Polozhi,*Equations of Mathematical Physics*, ``Vysshaya shkola'', Moscow, 1964. (Russian)

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Additional Information

**Yu. V. Kozachenko**

Affiliation:
Department of Probability Theory and Mathematical Statistics, Faculty for Mathematics and Mechanics, National Taras Shevchenko University, Glushkov Avenue, 6, Kyiv, 03127, Ukraine

Email:
yvk@univ.kiev.ua

**G. I. Slivka**

Affiliation:
Department of Mathematical Analysis, Faculty for Mathematics, Uzhgorod University, Pidgirna Street, 46, Uzhgorod, Ukraine

Email:
kafmatan@univ.uzhgorod.ua

DOI:
https://doi.org/10.1090/S0094-9000-07-00698-9

Received by editor(s):
February 14, 2005

Published electronically:
June 29, 2007

Article copyright:
© Copyright 2007
American Mathematical Society