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Theory of Probability and Mathematical Statistics

ISSN 1547-7363(online) ISSN 0094-9000(print)

 
 

 

Wave equation for a homogeneous string with fixed ends driven by a stable random noise


Authors: L. I. Rusanyuk and G. M. Shevchenko
Translated by: N. N. Semenov
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 98 (2018).
Journal: Theor. Probability and Math. Statist. 98 (2019), 171-181
MSC (2010): Primary 60H15, 35L05; Secondary 35R60, 60G52
DOI: https://doi.org/10.1090/tpms/1069
Published electronically: August 19, 2019
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Abstract: A wave equation with external forces is considered in this paper for a homogeneous string with fixed ends. The distribution of the right-hand side of the equation is symmetric $ \alpha $-stable. It is proved that the function constructed by the Fourier method is a generalized solution of the equation. The regularity of the trajectories is also established.


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Additional Information

L. I. Rusanyuk
Affiliation: Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, Kyiv Taras Shevchenko National University, Volodymyrs’ka Street, 64/13, Kyiv 01601, Ukraine
Email: pruhara7@gmail.com

G. M. Shevchenko
Affiliation: Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, Kyiv Taras Shevchenko National University, Volodymyrs’ka Street, 64/13, Kyiv 01601, Ukraine
Email: zhora@univ.kiev.ua

DOI: https://doi.org/10.1090/tpms/1069
Keywords: Wave equation for a string, wave equation, Fourier method, generalized solution, stable measure with independent increments, LePage representation
Received by editor(s): January 6, 2018
Published electronically: August 19, 2019
Article copyright: © Copyright 2019 American Mathematical Society