Abstract
We consider an inhomogeneous hyperbolic equation with zero initial and boundary conditions and a random centered sample-continuous Gaussian right-hand side. We establish conditions for the existence of a solution of the first boundary-value problem of mathematical physics in the form of a series uniformly convergent in probability in terms of a covariance function. An estimate for the distribution of the supremum of a solution of this problem is obtained.
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REFERENCES
V. B. Dovhai, “Justification of the Fourier method for an inhomogeneous hyperbolic equation with random right-hand side,” Ukr. Mat. Zh., 56, No.5, 616–624 (2004).
V. V. Buldygin and Yu. V. Kozachenko, “On the applicability of the Fourier method to the solution of problems with random boundary conditions,” in: Random Processes in Problems of Mathematical Physics [in Russian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1979), pp. 4–35.
V. V. Buldygin and Yu. V. Kozachenko, Metric Characterization of Random Variables and Random Processes, TViMS, Kiev (2000).
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Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 4, pp. 474–482, April, 2005.
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Dovhai, B.V. Properties of a Solution of an Inhomogeneous Hyperbolic Equation with Random Right-Hand Side. Ukr Math J 57, 571–582 (2005). https://doi.org/10.1007/s11253-005-0211-x
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DOI: https://doi.org/10.1007/s11253-005-0211-x