On the solution of the nonlinear Lippmann – Schwinger integral equation by the method of contracting maps 

Yuriy G. Smirnov, Doctor of physical and mathematical sciences, professor, head of the sub-department of mathematics and supercomputer modeling, Penza State University (40 Krasnaya street, Penza, Russia), E-mail: mmm@pnzgu.ru
Dar'ya A. Labutkina, Postgraduate student, Penza State University (40 Krasnaya street, Penza, Russia), E-mail: labutkinadara@gmail.com 

Background. A scalar three-dimensional boundary value problem of wave diffraction on an inhomogeneous scatterer for the Helmholtz equation with a nonlinear dependence of the scattering wavenumber on the field is considered. Materials and methods. The boundary value problem is reduced to the volume nonlinear Lippmann – Schwinger integral equation on the scatterer. The method of contracting maps is used to study the integral equation. Results. The existence and uniqueness of the solution in the space of continuous functions under certain conditions on the parameters of the problem are proved. Conclusions. The convergence of the iterative process in the method of contracting maps is proved and an estimate of the convergence rate is presented. 

Helmholtz equation, integral equations, method of contracting maps, solvability of boundary value problem, numerical method 

Smirnov Yu.G., Labutkina D.A. On the solution of the nonlinear Lippmann – Schwinger integral equation by the method of contracting maps. Izvestiya vysshikh uchebnykh zavedeniy. Povolzhskiy region. Fiziko-matematicheskie nauki = University proceedings. Volga region. Physical and mathematical sciences. 2023;(3):3–10. (In Russ.). doi:10.21685/2072-3040-2023-3-1