An iterative scheme for solving a Lippmann – Schwinger nonlinear integral equation by the Galerkin method 

Andrey O. Lapich, Assistant of the sub-department of mathematics and supercomputer modeling, Penza State University (40 Krasnaya street, Penza, Russia), E-mail: lapich.a@yandex.ru
Mikhail Yu. Medvedik, Candidate of physical and mathematical sciences, associate professor, associate professor of the sub-department of mathematics and supercomputer modeling, Penza State University (40 Krasnaya street, Penza, Russia), E-mail: _medv@mail.ru 

Background. The purpose of the work is to solve the nonlinear integral equation describing the propagation of electromagnetic waves inside a body located in free space. Materials and methods. The boundary value problem for the Helmholtz equation is reduced to the solution of the integral equation. An iterative method of creating a nonlinear medium inside the body with a dielectric structure is constructed. Results. The problem is solved numerically. The size of the matrix obtained in the calculation exceeds 30000 elements. The internal convergence of the iteration method is shown. The graphics illustrating the field distribution inside a nonlinear body are shown. Conclusions. A numerical method for finding the nonlinear field has been proposed and realized. 

boundary value problem, Helmholtz equation, Lippmann-Schwinger nonlinear volume integral equation, numerical method, Galerkin method 

Lapich A.O., Medvedik M.Yu. An iterative scheme for solving a Lippmann-Schwinger nonlinear inte-gral equation by the Galerkin method. Izvestiya vysshikh uchebnykh zavedeniy. Povolzhskiy region. Fiziko-matematicheskie nauki = University proceedings. Volga region. Physical and mathematical sciences. 2023;(3):66–73. (In Russ.). doi: 10.21685/2072-3040-2023-3-5