Nonlocal mixed-value problem for a Boussinesq-type integrodifferential equation with degenerate kernel
We consider the problem of one-valued solvability of the mixed-value problem for a nonlinear Boussinesq type fourth-order integrodifferential equation with degenerate kernel and integral conditions. The method of degenerate kernel is developed for the case of nonlinear Boussinesq type fourth-order p...
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| Datum: | 2016 |
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| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | Ukrainisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
2016
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| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/1906 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
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Ukrains’kyi Matematychnyi Zhurnal| Zusammenfassung: | We consider the problem of one-valued solvability of the mixed-value problem for a nonlinear Boussinesq type fourth-order
integrodifferential equation with degenerate kernel and integral conditions. The 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 employed. After redenoting, the integrodifferential equation is reduced to a system of countable system of
algebraic equations with nonlinear and complex right-hand side. 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 (CSNIE). To prove the theorem on one-valued solvability of the CSNIE, we use the method
of successive approximations. Further, we establish the convergence of the Fourier series to the required function of the
mixed-value problem. Our results can be regarded as a subsequent development of the theory of partial integrodifferential
equations with degenerate kernel. |
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