Nonlocal mixed-value problem for a Boussinesq-type integrodifferential equation with degenerate kernel
AbstractWe 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.
How to Cite
Yuldashev, T. K. “Nonlocal Mixed-Value Problem for a Boussinesq-Type integrodifferential equation With Degenerate Kernel”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 68, no. 8, Aug. 2016, pp. 1115-31, https://umj.imath.kiev.ua/index.php/umj/article/view/1906.