Journal of Applied Mathematics and Computation

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Article http://dx.doi.org/10.26855/jamc.2023.03.007

Mixed Problem for Inhomogeneous Wave Equation of Bounded String with Non-characteristic Second Derivatives in Non-stationary Boundary Modes

Lomovtsev Fedor Egorovich1, Lysenko Valery Vladimirovna2,*

1Doctor of Physical and Mathematical Sciences, Belarusian State University, Belarus, Minsk.

2Postgraduate student of Mathematical Cybernetics, Belarusian State University, Belarus, Minsk.

*Corresponding author: Lysenko Valery Vladimirovna

Published: April 12,2023

Abstract

It is found a classical solution to a mixed problem for an inhomogeneous wave equation of bounded string in the case of time-dependent coefficients and non-characteristic partial second order derivatives in boundary modes. A correctness criterion (necessary and sufficient conditions) according to Adamard (unique and stable everywhere solvability) is derived in the set of classical solutions without extensions of the problem data (the right-hand side of the equation, initial and boundary data) outside the sets of their specification. The correctness criterion for this problem includes the smoothness requirements and matching conditions on the mixed problem data. Therefore, the proved theorem is called the global correctness theorem. These results were obtained by Lomovtsev F.E., "method of auxiliary mixed problems for wave equations on the half-line”. First, the upper half-band of the plane is divided into rectangles of height equal to the passage time of the forward and backward waves. Then, by mathematical induction, the restrictions to these rectangles are taken of the previously established classical solution formula and correctness criterion for a similar non-characteristic mixed problem for a semi-bounded string.

References

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https://elib. bsu. by/handle/123456789/141737.

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How to cite this paper

Mixed Problem for Inhomogeneous Wave Equation of Bounded String with Non-characteristic Second Derivatives in Non-stationary Boundary Modes

How to cite this paper:  Lomovtsev Fedor Egorovich, Lysenko Valery Vladimirovna. (2023) Mixed Problem for Inhomogeneous Wave Equation of Bounded String with Non-characteristic Second Derivatives in Non-stationary Boundary Modes. Journal of Applied Mathematics and Computation7(1), 65-82.

DOI: http://dx.doi.org/10.26855/jamc.2023.03.007