Unique Solvability of an Initial–Boundary Value Problem for a System of Third-Order Partial Differential Equations

Abstract

We consider an initial–boundary value problem for a system of third-order partial differential equations in a rectangular domain. By introducing a new unknown function, we reduce the problem to an equivalent one consisting of a nonlocal problem for a system of second-order hyperbolic equations with parameters and integral relations. An iterative algorithm is proposed for approximately solving the equivalent problem, and its convergence is proved. Sufficient conditions in terms of the input data are established for the existence of a unique classical solution of the original problem.

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REFERENCES

  1. 1

    Ptashnyck, B.I., Nekorrektnye granichnye zadachi dlya differentsial’nykh uravnenii s chastnymi proizvodnymi (Ill-Posed Boundary Value Problems for Partial Differential Equations), Kiev: Nauk. Dumka, 1984.

    Google Scholar 

  2. 2

    Nakhushev, A.M., Uravneniya matematicheskoi biologii (Equations of Mathematical Biology), Moscow: Vyssh. Shkola, 1995.

    Google Scholar 

  3. 3

    Zhegalov, V.I. and Mironov, A.N., Differentsial’nye uravneniya so starshimi chastnymi proizvodnymi (Differential Equations with Leading Partial Derivatives), Kazan: Kazan. Mat. O-vo, 2001.

    Google Scholar 

  4. 4

    Nakhushev, A.M., Zadachi so smeshcheniem dlya uravnenii v chastnykh proizvodnykh (Problems with Displacement for Partial Differential Equations), Moscow: Nauka, 2006.

    Google Scholar 

  5. 5

    Asanova, A.T. and Dzhumabaev, D.S., Well-posedness of nonlocal boundary value problems with integral condition for the system of hyperbolic equations, J. Math. Anal. Appl., 2013, vol. 402, no. 1, pp. 167–178.

    MathSciNet  Article  Google Scholar 

  6. 6

    Assanova, A.T. and Imanchiev, A.E., On conditions of the solvability of nonlocal multi-point boundary value problems for quasi-linear systems of hyperbolic equations, Eurasian Math. J., 2015, vol. 6, no. 4, pp. 19–28.

    MathSciNet  MATH  Google Scholar 

  7. 7

    Asanova, A.T., Multipoint problem for a system of hyperbolic equations with mixed derivative, J. Math. Sci. (US), 2016, vol. 212, no. 3, pp. 213–233.

    MathSciNet  Article  Google Scholar 

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Funding

This work was supported by the Ministry of Education and Science of the Republic of Kazakhstan for 2020–2022, project no. AP08955461.

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Correspondence to A. T. Assanova.

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Translated by V. Potapchouck

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Assanova, A.T. Unique Solvability of an Initial–Boundary Value Problem for a System of Third-Order Partial Differential Equations. Diff Equat 57, 111–116 (2021). https://doi.org/10.1134/S0012266121010092

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