The Application of the Method of Two-Sided Approximations to Solving the Dirichlet Problem for a Semilinear Equation with a Biharmonic Operator, which is a Mathematical Model of a Microelectromechanical System

The article addresses the Dirichlet problem for a semilinear differential equation with a biharmonic operator, which arises in the mathematical modeling of various physical processes. Specifically, the study considers the deflection of a circular plate under the action of electrostatic force and hydrostatic pressure, modeling the operation of an actuator, the primary mechanism in most microelectromechanical systems (MEMS). The boundary value problem is solved using the method of two-sided approximations based on the method of Green's functions. This approach is chosen because it provides conditions for the existence of a solution to the original problem and offers a convenient a posteriori error estimation for the approximate solution. The solution process using the method of two-sided approximations is grounded in reducing the problem for a semilinear differential equation with a biharmonic operator to a nonlinear Hammerstein integral equation via the method of Green's functions. This equation was then analyzed using the theory of nonlinear operators in semi-ordered Banach spaces. An iterative process for finding a positive solution to the problem has been constructed, and conditions for its two-sided convergence have been established. Computational experiments for specific parameter values of the model were carried out and compared with the results obtained by other authors. Additionally, an analysis of the maximum deflection of the plate under varying pressure values and constant voltage was conducted. The novelty of this work lies in the fact that the method of two-sided approximations has been applied to a semilinear equation with a biharmonic operator for the first time. The results of the study demonstrate the successful application of the developed algorithm to model the considered process. Since the modeling focuses on an actuator consisting of circular plates, the obtained results can be utilized in the design and analysis of ultrasonic transducers, pressure sensors, miniature pumps, gas detectors, etc.