To model the moisture transport in soil and to better understand physics underneath, we study a boundary value problem for a nonlinear hyperbolic PDE. Using a constructive method for approximation of solutions of the problem, we derive sufficient conditions for existence and uniqueness of its regular solutions and show that these solutions satisfy the sign-preserving inequalities. Additionally, we prove a comparison theorem and a theorem about differential inequalities, and derive an posteriori error of the method. Theoretical results are validated on an illustrative numerical example.

We study a boundary value problem for a system of the third order semi-linear partial differential equations with nonlocal boundary conditions. We establish sufficient conditions of existence, uniqueness, regularity and sign-preserving property of solutions of the studied problem and construct an iterative method for its approximation.

We construct the modification of the two-sided method with improved characteristics of convergence for approximate integration of boundary-value problem for systems of secondorder non-linear differential equations of the hyperbolic type on the plane when the bound of the domain of the independent variables consists of the pair of characteristics.