Application of the continuous operator method to the solution of the Pocklington and Gallen equations for thin wire antennas

Background. One of the central tasks in microwave electronics is the construction of miniature antennas with high performance. The main equations used in modeling wire antennas of various configurations are the Pocklington, Gallen equations, singular and hypersingular integral equations. In the numerical solution of the Pocklington and Gallen equations, the methods of moments and Galerkin are mainly used. Since the Pocklington and Gallen equations belong to the class of ill-posed problems, when implementing the methods of moments and Galerkin, additional difficulties arise due to the instability of computational schemes. In this paper, to solve the Pocklington and Gallen equations, it is proposed to apply a continuous method for solving operator equations, which has the effect of regularization. This effect is due to the fact that the continuous method for solving operator equations is based on the Lyapunov theory of stability of solutions of differential equations.
Materials and methods. The article investigates approximate methods for solving the Pocklington and Gallen equations. The method for constructing computational schemes is as follows. The initial equations are approximated by a system of linear algebraic equations constructed using the spline collocation method. The system of linear algebraic equations is solved by a continuous operator method. The advantages of the proposed method should be noted: 1) its stability under perturbation of the kernels of equations and right-hand sides; 2) the construction of a computational scheme allows taking into account the boundary conditions at the ends of the vibrator.
Results. New stable numerical methods for solving the Pocklington and Gallen equations are constructed. The effectiveness of the proposed methods is demonstrated by solving model examples.
Conclusions. The proposed method for constructing and justifying the convergence of computational schemes can be extended to equations that model various antenna modifications and are analogous to equations of the Pocklington and Gallen type.