NECESSARY CONDITIONS FOR EXISTENCE OF CLASSICAL SOLUTIONS TO THE EQUATION OF SEMI-BOUNDED STRING VIBRATION
Abstract
If the inhomogeneous equation of semi-bounded string vibration is a classical solution in the first quadrant, then the right-hand side of this equation is obviously continuous. We prove that in this case, a special integral of this right-hand side, which is a generalized solution of the inhomogeneous equation for semi-bounded string vibration, has continuous second derivatives and it is therefore a classical solution. This generalized solution differs from the known generalized solution of this equation in the presence of the upper half of the module of the spatial variable in the integrand, which is a continuous right-hand side of
the equation. This assertion can be used to identify the corresponding necessary smoothness requirements on the right-hand side of the equation for string vibration for the existence of classical solutions of different mixed problems in the quarter and the half-plane.
the equation. This assertion can be used to identify the corresponding necessary smoothness requirements on the right-hand side of the equation for string vibration for the existence of classical solutions of different mixed problems in the quarter and the half-plane.
About the Authors
N. I. Yurchuk
Belarusian State University
Belarus
D. Sc. (Physics and Mathematics), Professor of the Department of Mathematical Cybernetics of the Faculty of Mechanics and Mathematics
Belarus
D. Sc. (Physics and Mathematics), Professor of the Department of Mathematical Cybernetics of the Faculty of Mechanics and Mathematics
E. N. Novikov
Belarusian State University
Belarus
Postgraduate
Belarus
Postgraduate
References
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