## 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.

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

[1] Lomovtsev, F. E. (2015). Method of auxiliary mixed problems for a semi-limited string/Proceedings of the International Mathematical Conference The Sixth Bogdanov Readings on Ordinary Differential Equations. December 7-10, 2015. 74-75.

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

[2] Lomovtsev, F. E., Lysenko V. V. (2019). Uncharacteristic mixed problem for a one-dimensional wave equation in the first quarter of the plane at non-stationary boundary second derivatives. Bulletin of Vitebsk State University, 3 (104): 5-17.

https://rep. vsu. by/handle/123456789/19086.

[3] Lomovtsev, F. E. (2017). Correction method of test solutions to the general wave equation in the first quarter of the plane for the minimum smoothness of its right-hand side. Journal of the Belarusian State University. Mathematics. Computer science, 3: 38-52.

https://www. mathnet. RU/RUS/BGUMI143.

[4] Lomovtsev, F. E., Sholomitskaya, V. V. (2016). Mixed problem for inhomogeneous equation of a semi-bounded string oscillation at non-stationary first oblique and second derivative on x in boundary mode. Bulletin of the Belarusian State University, series 1, 2: 95-102.

https://elib. bsu. by/handle/123456789/172056.

[5] Moiseev, E. I., .Lomovtsev, F. E., Novikov, E. N. (2014). Inhomogeneous factorized hyperbolic equation of the second order in a quarter of the plane at a semi-non-stationary factorized second oblique derivative in the boundary condition. Reports of the Academy of Sciences, vol. 459, 5: 544-549.

[6] Novikov, E. N. (2017). Mixed problems for the forced oscillation equation of a bounded string under non-stationary boundary modes with the first and second oblique derivatives. Ph.D. Dissertation. Belarus Mathematical Institute of National Academy of Sciences. Minsk.

[7] Lomovtsev, F. E., Spesivtseva, K. A. (2021). Mixed Problem for a General 1D Wave Equation with Characteristic Second Derivatives in a Non-Stationary Boundary Mode. Mathematical notes, vol. 1, p. 110, 3: 329-338.

[8] Tikhonov, A. A., Samarsky А. А. (2004). Equations of Mathematical Physics. Nauka, Moscow: p. 798.

[9] Lomovtsev, F. Е. (2016). On global theorems with explicit solutions and correctness conditions of initial-boundary problems for the oscillation equation of a bounded string. / Materials of the International Conference. Voronezh Winter School of Mathematics by S. G. Krein - 2016. January 25-31, 2016. 279-282.

## 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