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Nonlocal Mixed-Value Problem for a Boussinesq-Type Integrodifferential Equation with Degenerate Kernel

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Ukrainian Mathematical Journal Aims and scope

We consider the problem of unique solvability of a mixed problem for a nonlinear Boussinesq-type fourthorder integrodifferential equation with degenerate kernel and integral conditions. A method of degenerate kernel is developed for the case of nonlinear Boussinesq-type fourth-order partial integrodifferential equation. The Fourier method of separation of variables is used. After redenoting, the integrodifferential equation is reduced to a system of countable systems of algebraic equations with nonlinear and complex right-hand sides. As a result of the solution of this system of countable systems of algebraic equations and substitution of the obtained solution in the previous formula, we get a countable system of nonlinear integral equations. To prove the theorem on unique solvability of the countable system of nonlinear integral equations, we use the method of successive approximations. Further, we establish the convergence of the Fourier series to the required function of the mixed problem. Our results can be regarded as a subsequent development of the theory of partial integrodifferential equations with degenerate kernels.

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Authors and Affiliations

  1. Siberian State Aerospace University, Krasnoyarsk, Russia

    T. K. Yuldashev

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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 68, No. 8, pp. 1115–1131, August, 2016.

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Yuldashev, T.K. Nonlocal Mixed-Value Problem for a Boussinesq-Type Integrodifferential Equation with Degenerate Kernel. Ukr Math J 68, 1278–1296 (2017). https://doi.org/10.1007/s11253-017-1293-y

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  • DOI: https://doi.org/10.1007/s11253-017-1293-y

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