The Method of Two-Sided Approximations in the Numerical Analysis of the Navier Problem as a Mathematical Model of a One-Dimensional Microelectromechanical System

• Datum: 2026
• Hauptverfasser: Гвоздєв, Микита, Сидоров, Максим
• Format: Artikel
• Sprache: Ukrainisch
• Veröffentlicht: Кам'янець-Подільський національний університет імені Івана Огієнка 2026
• Online Zugang: https://mcm-math.kpnu.edu.ua/article/view/362142
• Назва журналу: Mathematical and computer modelling. Series: Physical and mathematical sciences

This paper considers a boundary value problem for a fourth-order semilinear ordinary differential equation (the Navier problem), which describes the static deflection of a microbeam in microelectromechanical systems under the action of electrostatic forces. The study of this problem is based on its reduction to an equivalent Hammerstein integral equation or to a system of Hammerstein integral equations, which are analyzed using methods of nonlinear operator theory in semi-ordered Banach spaces. By the first approach, the original boundary value problem is reduced to a Hammerstein integral equation via the construction of the Green’s function for a fourth-order ordinary differential operator with Navier boundary conditions. In the second approach, the problem is first transformed into a first boundary value problem for a system of semilinear ordinary differential equations, followed by its reduction to a system of Hammerstein equations. In this case, Green’s functions of second-order ordinary differential operators with first boundary conditions are employed. The properties of the nonlinear operators corresponding to the obtained equation and system of equations are investigated. In particular, it is established that each of these operators is positive, isotone, Lipschitz continuous, continuous, and completely continuous. Two schemes of the method of two-sided approximations are proposed. The choice of this method is justified by the fact that it allows not only the construction of approximate solutions but also the theoretical establishment of conditions for their existence and uniqueness. Another advantage of the method is the availability of a convenient a posteriori error estimate. Conditions for the convergence of each of the proposed schemes to the unique solution of the original boundary value problem on an invariant conical segment are obtained. To analyze the efficiency of the algorithms, a series of computational experiments is carried out for different values of the problem parameters. A comparative analysis of the results is performed, and practical recommendations are provided.