Fractional solutions of Helmholtz equation in scattering problems

2004 ◽  
Author(s):  
T.M. Ahmedov ◽  
M.V. Ivakhnychenko ◽  
E.I. Veliev
Keyword(s):  
Helmholtz Equation ◽  
Scattering Problems

Novel solutions of the Helmholtz equation and their application to diffraction

2007 ◽  
Vol 463 (2080) ◽  
pp. 1005-1027 ◽  
Author(s):  
Bair V Budaev ◽  
David B Bogy
Keyword(s):  
Differential Equations ◽  
Helmholtz Equation ◽  
Broad Class ◽  
Probabilistic Approach ◽  
Three Dimensional ◽  
Boundary Function ◽  
Original Method ◽  
Random Motion ◽  
Scattering Problems ◽  
Homogeneous Media

This paper presents a series of novel representations for the solutions of the Helmholtz equation in a broad class of wedge-like domains including those with curvilinear, non-flat faces. These representations are obtained by an original method which combines ray theory with the probabilistic approach to partial differential equations and uses a specific technique to deal with a need for analytical continuation of the specified boundary function. The main results are reminiscent of the standard Feynman–Kac formula but differ in that the averaging over solutions of stochastic differential equations is replaced by averaging over the trajectories of a new two-scale random motion introduced here. The paper focuses on the development of the solutions and for this reason it includes only a brief outline of numerous applications, consequences, extensions and variations of the method, which include, but are not limited to, problems of diffraction and scattering, problems in three-dimensional domains and problems of wave propagation in non-homogeneous media.

Direct and inverse scalar scattering problems for the Helmholtz equation in ℝm

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Yury G. Smirnov ◽  
Aleksei A. Tsupak
Keyword(s):  
Integral Equation ◽  
Helmholtz Equation ◽  
Step Method ◽  
Scattering Problems ◽  
Inverse Coefficient Problem ◽  
Function Classes ◽  
Source Type ◽  
Scalar Scattering ◽  
Uniqueness Of A Solution ◽  
Inverse Coefficient

AbstractThe boundary value problem for the Helmholtz equation in the m-dimensional free space is considered. The problem is reduced to the Lippmann–Schwinger integral equation over the inhomogeneity domain. The operator of the integral equation is shown to be an invertible Fredholm operator. The inverse coefficient problem is considered. An application of the two-step method reduces the inverse problem to the source-type integral equation with a smooth kernel. Special classes of solutions to this equation are introduced. The uniqueness of a solution to the integral equation of the first kind is proved in the defined function classes.

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 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.

Domain decomposition preconditioners for high-order discretizations of the heterogeneous Helmholtz equation

2020 ◽  
Author(s):  
Shihua Gong ◽  
Ivan G Graham ◽  
Euan A Spence
Keyword(s):  
Helmholtz Equation ◽  
Domain Decomposition ◽  
Heterogeneous Media ◽  
Local Variation ◽  
Variable Coefficients ◽  
Impedance Boundary Condition ◽  
Scattering Problems ◽  
Impedance Boundary ◽  
Polynomial Degree ◽  
Field Of Values

Abstract We consider one-level additive Schwarz domain decomposition preconditioners for the Helmholtz equation with variable coefficients (modelling wave propagation in heterogeneous media), subject to boundary conditions that include wave scattering problems. Absorption is included as a parameter in the problem. This problem is discretized using $H^1$-conforming nodal finite elements of fixed local degree $p$ on meshes with diameter $h = h(k)$, chosen so that the error remains bounded with increasing $k$. The action of the one-level preconditioner consists of the parallel solution of problems on subdomains (which can be of general geometry), each equipped with an impedance boundary condition. We prove rigorous estimates on the norm and field of values of the left- or right-preconditioned matrix that show explicitly how the absorption, the heterogeneity in the coefficients and the dependence on the degree enter the estimates. These estimates prove rigorously that, with enough absorption and for $k$ large enough, GMRES is guaranteed to converge in a number of iterations that is independent of $k,p$ and the coefficients. The theoretical threshold for $k$ to be large enough depends on $p$ and on the local variation of coefficients in subdomains (and not globally). Extensive numerical experiments are given for both the absorptive and the propagative cases; in the latter case, we investigate examples both when the coefficients are nontrapping and when they are trapping. These experiments support (i) our theory in terms of dependence on polynomial degree and the coefficients; and (ii) the sharpness of our field of values estimates in terms of the level of absorption required.

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