# Nonlocal mixed-value problem for a Boussinesq-type integrodifferential equation with degenerate kernel

### Abstract

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.
Published

25.08.2016

How to Cite

*Ukrains’kyi Matematychnyi Zhurnal*, Vol. 68, no. 8, Aug. 2016, pp. 1115-31, http://umj.imath.kiev.ua/index.php/umj/article/view/1906.

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Section

Research articles