scholarly journals Continuous operator method application for direct and inverse scattering problems

2021 ◽  
Vol 23 (3) ◽  
pp. 247-272
Author(s):  
Ilya V. Boykov ◽  
Vladimir A. Roudnev ◽  
Alla I. Boykova ◽  
Nikita S. Stepanov
Keyword(s):  
Helmholtz Equation ◽  
Integral Equations ◽  
Inverse Scattering ◽  
Nonlinear Operator ◽  
Operator Method ◽  
Continuous Operator ◽  
Wave Number ◽  
Operator Equations ◽  
Scattering Problems ◽  
Neumann Problems

Abstract. We describe the continuous operator method for solution nonlinear operator equations and discuss its application for investigating direct and inverse scattering problems. The continuous operator method is based on the Lyapunov theory stability of solutions of ordinary differential equations systems. It is applicable to operator equations in Banach spaces, including in cases when the Frechet (Gateaux) derivative of a nonlinear operator is irreversible in a neighborhood of the initial value. In this paper, it is applied to the solution of the Dirichlet and Neumann problems for the Helmholtz equation and to determine the wave number in the inverse problem. The internal and external problems of Dirichlet and Neumann are considered. The Helmholtz equation is considered in domains with smooth and piecewise smooth boundaries. In the case when the Helmholtz equation is considered in domains with smooth boundaries, the existence and uniqueness of the solution follows from the classical potential theory. When solving the Helmholtz equation in domains with piecewise smooth boundaries, the Wiener regularization is carried out. The Dirichlet and Neumann problems for the Helmholtz equation are transformed by methods of potential theory into singular integral equations of the second kind and hypersingular integral equations of the first kind. For an approximate solution of singular and hypersingular integral equations, computational schemes of collocation and mechanical quadrature methods are constructed and substantiated. The features of the continuous method are illustrated with solving boundary problems for the Helmholtz equation. Approximate methods for reconstructing the wave number in the Helmholtz equation are considered.

scholarly journals Solvability and Bifurcation of Solutions of Nonlinear Equations with Fredholm Operator

Symmetry ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 912
Author(s):  
Nikolai Sidorov ◽  
Denis Sidorov ◽  
Aliona Dreglea
Keyword(s):  
Integral Equations ◽  
Nonlinear Operator ◽  
Sufficient Conditions ◽  
Branch Points ◽  
Operator Equations ◽  
Nonlinear Operator Equations ◽  
Nonlinear Functional ◽  
Branching Points ◽  
Nonstandard Models ◽  
Necessary And Sufficient

The necessary and sufficient conditions of existence of the nonlinear operator equations’ branches of solutions in the neighbourhood of branching points are derived. The approach is based on the reduction of the nonlinear operator equations to finite-dimensional problems. Methods of nonlinear functional analysis, integral equations, spectral theory based on index of Kronecker-Poincaré, Morse-Conley index, power geometry and other methods are employed. Proposed methodology enables justification of the theorems on existence of bifurcation points and bifurcation sets in the nonstandard models. Formulated theorems are constructive. For a certain smoothness of the nonlinear operator, the asymptotic behaviour of the solutions is analysed in the neighbourhood of the branch points and uniformly converging iterative schemes with a choice of the uniformization parameter enables the comprehensive analysis of the problems details. General theorems and effectiveness of the proposed methods are illustrated on the nonlinear integral equations.

Numerical solution of the system of Marchenko integral equations

2021 ◽  
Vol 65 (3) ◽  
pp. 159-165
Keyword(s):  
Differential Equations ◽  
Integral Equations ◽  
Inverse Scattering ◽  
Scattering Problem ◽  
Inverse Scattering Problem ◽  
Algebraic Equations ◽  
System Of Differential Equations ◽  
Scattering Problems ◽  
First Order ◽  
First Order Differential Equations

In this paper, inverse scattering problems for a system of differential equations of the first order are considered. The Marchenko approach is used to solve the inverse scattering problem. The system of Marchenko integral equations is reduced to a linear system of algebraic equations such that the solution of the resulting system yields to the unknown coefficients of the system of first-order differential equations. Illustrative examples are provided to demonstrate the preciseness and effectiveness of the proposed technique. The results are compared with the exact solution by using computer simulations.

scholarly journals Solvability of Nonlinear Equations in Case of Branching Solutions

2020 ◽  
Author(s):  
Nikolai A. Sidorov ◽  
Denis Sidorov ◽  
Aliona Dreglea
Keyword(s):  
Integral Equations ◽  
Nonlinear Operator ◽  
Sufficient Conditions ◽  
Branch Points ◽  
Operator Equations ◽  
Nonlinear Operator Equations ◽  
Nonlinear Functional ◽  
Branching Points ◽  
Nonstandard Models ◽  
Necessary And Sufficient

The necessary and sufficient conditions of existence of the nonlinear operator equations' branches of solutions in the neighbourhood of branching points are derived. The approach is based on reduction of the nonlinear operator equations to finite-dimensional problems. Methods of nonlinear functional analysis, integral equations, spectral theory based on index of Kronecker-Poincare, Morse-Conley index, power geometry and other methods are employed. Proposed methodology enables justification of the theorems on existence of bifurcation points and bifurcation sets in the nonstandard models. Formulated theorems are constructive. For a certain smoothness of the nonlinear operator, the asymptotic behaviour of the solutions is analysed in the neighbourhood of the branch points and uniformly converging iterative schemes with a choice of the uniformization parameter enables the comprehensive analysis of the problems details. General theorems are illustrated on the nonlinear integral equations.

scholarly journals Asymptotic solutions of the Helmholtz equation: Generalised Friedlander–Keller ray expansions of fractional order

2018 ◽  
Vol 31 (1) ◽  
pp. 1-25 ◽  
Author(s):  
R. H. TEW
Keyword(s):  
Helmholtz Equation ◽  
Boundary Value ◽  
Fractional Power ◽  
Boundary Curve ◽  
Asymptotic Solutions ◽  
Wave Number ◽  
Small Scale ◽  
Scattering Problems ◽  
Large Wave ◽  
Exponential Factor

Applications of a WKBJ-type ‘ray ansatz’ to obtain asymptotic solutions of the Helmholtz equation in the high-frequency limit are now standard and underpin the construction of ‘geometrical optics’ ray diagrams in many electromagnetic, acoustic and elastic reflection, transmission and other scattering problems. These applications were subsequently extended by Keller to include other types of rays – called ‘diffracted’ rays – to provide an accessible and impressively accurate theory which is relevant in wide-ranging sets of circumstances. Friedlander and Keller then introduced a modified ray ansatz to extend yet further the scope of ray theory and its applicability to certain other classes of diffraction problems (tangential ray incidence upon an obstructing boundary, for instance) and did so by the inclusion of an extra term proportional to a power of the wave number within the exponent of the initial ansatz. Our purpose here is to generalise this further still by the inclusion of several such terms, ordered in a natural sequence in terms of strategically chosen fractional powers of the large wave number, and to derive a systematic sequence of boundary value problems for the coefficient phase functions that arise within this generalised exponent, as well as one for the leading-order amplitude occurring as a pre-exponential factor. One particular choice of fractional power is considered in detail, and waves with specified radially symmetric or planar wavefronts are then analysed, along with a boundary value problem typifying two-dimensional radiation whereby arbitrary phase and amplitude variations are specified on a prescribed boundary curve. This theory is then applied to the scattering of plane and cylindrical waves at curved boundaries with small-scale perturbations to their underlying profile.

Sign in / Sign up

Export Citation Format

Share Document