Justification of the Fourier method for hyperbolic equations with random initial conditions
Authors:
Yu. V. Kozachenko and G. I. Slivka
Translated by:
Oleg Klesov
Journal:
Theor. Probability and Math. Statist. 69 (2004), 67-83
MSC (2000):
Primary 60G35; Secondary 35L20
DOI:
https://doi.org/10.1090/S0094-9000-05-00615-0
Published electronically:
February 8, 2005
MathSciNet review:
2110906
Full-text PDF Free Access
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Abstract: Conditions for the existence of a twice differentiable solution of a hyperbolic type partial differential equation with random strongly $\operatorname {Sub}_{\varphi }(\Omega )$ initial conditions are found in the multidimensional case.
References
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- M. A. Krasnosel′skiĭ and Ja. B. Rutickiĭ, Vypuklye funktsii i prostranstva Orlicha, Problems of Contemporary Mathematics, Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, 1958 (Russian). MR 0106412
- Yu. V. Kozachenko and Yu. A. Koval′chuk, Boundary value problems with random initial conditions, and functional series from ${\rm sub}_\phi (\Omega )$. I, Ukraïn. Mat. Zh. 50 (1998), no. 4, 504–515 (Russian, with English and Ukrainian summaries); English transl., Ukrainian Math. J. 50 (1998), no. 4, 572–585 (1999). MR 1698149, DOI https://doi.org/10.1007/BF02487389
- O. N. Gladkaja, Conditions for the directional differentiability of sample functions of random fields, Teor. Verojatnost. i Mat. Statist. Vyp. 17 (1977), 33–41, 164 (Russian, with English summary). MR 0451375
- Yu. Kozachenko, O. Vasylyk, and T. Sottinen, Path space large deviations of a large buffer with Gaussian input traffic, Queueing Syst. 42 (2002), no. 2, 113–129. MR 1932123, DOI https://doi.org/10.1023/A%3A1020161203956
- N. S. Košljakov, È. B. Gliner, and M. M. Smirnov, Differentsial′nye uravneniya matematicheskoĭ fiziki, Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, 1962 (Russian). Edited by N. S. Košljakov. MR 0158145
- Yu. V. Kozachenko, Local properties of trajectories of some random functions, Ukr. Matem. Zh. 17 (1977), 33–40. (Russian)
- Ganna Ī. Slivka, Justification of the application of the Fourier method to the problem of the vibration of a round membrane with random initial conditions, Vīsn. Kiïv. Unīv. Ser. Fīz.-Mat. Nauki 4 (2002), 31–37 (Ukrainian, with English and Ukrainian summaries). MR 1972945
- G. I. Slivka, A boundary value problem of the mathematical physics with random initial conditions, Visn. Kyïv Univ. Ser. Fiz.-Mat. Nauk 5 (2002), 172–178. (Ukrainian)
References
- V. V. Buldygin and Yu. V. Kozachenko, Metric Characterization of Random Variables and Random Processes, Amer. Math. Soc., Providence, RI, 2000. MR 1743716 (2001g:60089)
- M. A. Krasnosel’skiĭ and Ya. B. Rutickiĭ, Convex Functions and Orlicz Spaces, Fizmatgiz, Moscow, 1958; English transl., Noordhof, Gröningen, 1961. MR 0106412 (21:5144)
- Yu. V. Kozachenko and Yu. A. Koval’chuk, Boundary value problems with random initial conditions and series of functions of $\Sub _\varphi (\Omega )$, Ukr. Matem. Zh. 50 (1998), no. 4, 504–515; English transl. in Ukrainian Math. J. 50 (1998), no. 4, 572–585 (1999). MR 1698149 (2000f:60029)
- O. N. Gladkaya, Conditions for directional differentiability of the sample functions of random fields, Teor. Veroyatnost. Matem. Statist. 17 (1977), 33–40; English transl. in Theory Probab. Mathem. Statist. 17 (1977), 35–43. MR 0451375 (56:9661)
- Yu. V. Kozachenko, O. Vasylyk, and T. Sottinen, Path space large deviations of a large buffer with Gaussian input traffic, Queueing System 42 (2002), no. 2, 113–129. MR 1932123 (2004a:60149)
- N. S. Koshlyakov, E. B. Gliner, and M. M. Smirnov, Differential Equations of Mathematical Physics, “Vysshaya Shkola”, Moscow, 1962; English transl., North-Holland, Amsterdam, 1964. MR 0158145 (28:1371)
- Yu. V. Kozachenko, Local properties of trajectories of some random functions, Ukr. Matem. Zh. 17 (1977), 33–40. (Russian)
- G. I. Slivka, Justification of the application of the Fourier method to the problem of the vibration of a round membrane with random initial conditions, Visn. Kyïv Univ. Ser. Fiz.-Mat. Nauk 4 (2002), 31–37. (Ukrainian) MR 1972945
- G. I. Slivka, A boundary value problem of the mathematical physics with random initial conditions, Visn. Kyïv Univ. Ser. Fiz.-Mat. Nauk 5 (2002), 172–178. (Ukrainian)
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Additional Information
Yu. V. Kozachenko
Affiliation:
Department of Probability Theory and Mathematical Statistics, Faculty for Mechanics and Mathematics, Kyiv National Taras Shevchenko University, Academician 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:
aslyvka@yahoo.com
Received by editor(s):
March 28, 2003
Published electronically:
February 8, 2005
Article copyright:
© Copyright 2005
American Mathematical Society
