Yuldashev T. K.
Nonlocal mixed-value problem for a Boussinesq-type integrodifferential equation with degenerate kernel
Ukr. Mat. Zh. - 2016. - 68, № 8. - pp. 1115-1131
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.
Volterra functional integral equation of the first kind with nonlinear right-hand side and variable limits of integration
Ukr. Mat. Zh. - 2006. - 58, № 9. - pp. 1285–1288
We prove a theorem on the existence and uniqueness of a solution of a Volterra functional integral equation of the first kind with nonlinear right-hand side and nonlinear deviation. We use the method of successive approximations combined with the method of contracting mappings.