Advances in Mathematical Physics

Volume 2015, Article ID 945965, 6 pages

http://dx.doi.org/10.1155/2015/945965

## Integrodifferential Equations of the Vector Problem of Electromagnetic Wave Diffraction by a System of Nonintersecting Screens and Inhomogeneous Bodies

^{1}Department of Mathematics and Supercomputer Modeling, Penza State University, Penza 440026, Russia^{2}Department of Mathematics, Penza State University, Krasnaya Street 40, Penza 440026, Russia

Received 20 November 2014; Revised 8 April 2015; Accepted 15 April 2015

Academic Editor: Prabir Daripa

Copyright © 2015 Y. G. Smirnov and A. A. Tsupak. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The vector problem of electromagnetic wave diffraction by a system of bodies and infinitely thin screens is considered in a quasi-classical formulation. The solution is sought in the classical sense but is defined not in the entire space but rather everywhere except for the screen edges. The original boundary value problem for Maxwell’s equations system is reduced to a system of integrodifferential equations in the regions occupied by the bodies and on the screen surfaces. The integrodifferential operator is treated as a pseudodifferential operator in Sobolev spaces and is shown to be zero-index Fredholm operator.

#### 1. Introduction

In this work, we obtain new integrodifferential equations of the vector problem of electromagnetic wave diffraction by a set of two- and three-dimensional scatterers of complex geometry.

Acoustic and electromagnetic problems of diffraction by screens or inhomogeneous bodies are classical ones, which have been thoroughly investigated by many researchers. In [1], a wide class of acoustic scattering problems in bounded and unbounded regions is described. In [2, 3], boundary value problems (or transmission problems) are reduced to surface integral equations over the boundary of a region, followed by their analysis and numerical solution by applying projection methods. Scalar problems of diffraction by unclosed screens and their numerical solution are described, for example, in [4].

Methods of volume integral equations (as well as integrodifferential equations) are also applied for theoretical study and numerical solving of electromagnetic problems (see [5, 6] and the bibliography inside these works).

Diffraction problems (mostly, vector ones) are the subject of a number of works by the authors of this paper. In [7], vector problems of diffraction by thin conducting screens are studied by applying methods of the theory of pseudodifferential operators. In [8], diffraction problems were investigated both theoretically (the existence and uniqueness of boundary value problems, their equivalence to integral equations, and the invertibility of integral operators were considered) and numerically (Galerkin method was formulated and its convergence was proved). In [9], methods of pseudodifferential operators were applied to prove ellipticity and Fredholm property of volume integrodifferential operator.

In the present work, we consider a more complicated problem: the scattering structure is formed of three-dimensional bodies and infinitely thin screens. Such a problem (namely, the scalar problem of diffraction by bodies and surfaces) was treated first in [10]. Both in [10] and in this paper, the solution is understood as a function defined everywhere, except for the edge of the screens; it satisfies Maxwell’s equations in the classical sense, continuity conditions on the interface between the media, Dirichlet condition in the interior points of the screens, and radiation conditions at infinity. For this problem, we prove a uniqueness theorem.

Below, the original boundary value problem is reduced to a system of integrodifferential equations. According to well-known results on the behavior of the field near a screen edge, it is necessary to consider the equations obtained as pseudodifferential equations in Sobolev spaces. Using this approach we prove that the solution is smooth at points other than the screen edges and establish the Fredholm property of the matrix integrodifferential operator of the considered diffraction problem.

#### 2. The Statement of the Diffraction Problem

Let be the union of a finite number of connected oriented unclosed and nonintersecting bounded surfaces of class in . The edge of the surface consists of a finite number of curves of the class without self-intersection points that intersect at nonzero angles and . We consider all screens to be infinitely conducting.

Define tubular neighborhoods of the screen edge as

Assume that are bounded regions whose boundaries are piecewise smooth closed oriented surfaces consisting of a finite number of surfaces of the class . Let . Assume that . The bodies are anisotropic and inhomogeneous. The inhomogeneity of the problem is described by the tensor function where . Hereinafter, is the unit tensor; ; the free space is isotropic and homogeneous with constant and . In the inhomogeneity area , the condition holds. We assume that are symmetric complex tensors with nonnegative imaginary part:

The conditions , , , and are satisfied in the free space .

We consider the problem of diffraction of electromagnetic wave , depending on time harmonically as by a system of bodies and infinitely thin screens.

The incident field is the solution to Maxwell’s equations in homogeneous entire space:Here current has a compact support such that .

We also define , as arbitrary domains exterior and interior with respect to the screen assuming that .

The diffraction problem consists in finding a total electromagnetic field :satisfying Maxwell’s equationsoutside the screens and the boundaries of the bodies, the transmission conditionson the boundary of the inhomogeneous bodies, the Dirichlet boundary conditionsin the interior points of the screen (i.e., everywhere on except for the edge points), the energy finiteness conditionin any bounded space volume, and the the Silver-Mueller radiation conditions at infinityfor the scattered field , ( and ).

*Definition 1. *The solution of problem (7)–(11) satisfying condition (6) is called quasi-classical.

Theorem 2. *The inhomogeneous boundary value problem (7)–(11) has at most one quasi-classical solution.*

It suffices to show that the homogeneous boundary value problem for the scattered field has only the trivial solution.

The boundary value problem for is formulated as follows:The Silver-Mueller conditions should also be satisfied.

The screen is extended to an arbitrary piecewise smooth closed simply connected orientable surface surrounding a bounded region such that .

Let be a ball of a sufficiently large radius such that

Let . Define the region with the boundary , and consider .

Henceforward, denotes the derivative in the direction of the unit outward normal vector to the domain of and denote the scattered field as considered in the region .

The boundary value problem for the scattered field is reduced to the following transmission problem in the regions :

We will apply the integral Lorentz lemmain the bounded regions , , and .

In the region , consider Maxwell’s equationsfor the scattered field as well as for the conjugate field :Replacing by , we obtainApplying the Lorentz lemma to the fields , , , and and currents , , we derive the following equalityReplacing with and taking into account the media properties we obtain

In the regions and , the similar integral relations hold

Define

Since the Silver-Mueller conditions hold, we obtain

Sum equalities (20), (21), and (22):Consider several cases.

Suppose that everywhere in . Then radiation conditions imply in the entire space from (25).

If in and in the free space, then we have As both terms in the upper relation are nonnegative, we obtain By the Rellich lemma, we conclude that outside the scatterers; the second equation implies in .

Finally, if permittivity is everywhere in real and positive we derive by the Rellich lemma that outside the scatterers. In paper [6] it is proved that holds in .

#### 3. Integrodifferential Equations

Represent the total field as the sumwhere is the incident field and is the field scattered by in the absence of . Hence, we have .

As mentioned above, satisfies Maxwell’s equations as well as transmission conditions

The field is a solution to Maxwell’s equations in :satisfying radiation conditions (11) as well as transmission conditions

The field is sought in the following form:where , is unknown density of the surface current on , and for all vector functions . We assume that is a* tangential* vector field: on ( denotes unit normal vector field on ).

Define “new” incident field and write system (7) as follows:Here current is defined as where are the currents corresponding to the field and is the polarization current in the inhomogeneity domain :

In we define electric field via vector potentialaccording to well-known ([11], page 61) formulas

Thus definition of the fields and as well as equalities (28), (32), (36), and (37) implies the following integrodifferential equation:

In a similar manner, the field outside the bodies and the screens is represented as

To obtain the second equation we use condition (9). Placing the point on in (39) and considering the tangential components of all terms we have

Define and and write system (39), (40) for currents:

Define the matrix operator of system (41):Here operators , , and are defined according to (41):and they are regarded as mappings in the spaces:

The space of vector bundles over was introduced in [7] as the closure of with respect to the norm : Here denotes the norm in Sobolev space and is the antidual space to (see [7], page 88).

The solution to the system of integrodifferential equations is understood as a pair of functions , where is the polarization current corresponding to the total electric field in and is the density of the surface current on .

Theorem 3. *In the spaces chosen, is a Fredholm operator.*

Throughout the following, a Fredholm operator is treated as a zero-index Noether operator.

As the permittivity is a positive function in then the operator is a Fredholm operator in (see [9]). The operator is a Fredholm operator from to because ([7], page 109). Therefore, is a Fredholm operator.

The operator is compact because the kernels of the operators and are infinitely smooth.

Finally, is a Fredholm operator in the above-mentioned spaces.

*Statement 1. *If solution exists, then and are infinitely differentiable in the interior points of and , respectively.

Suppose that is a solution to system (41) with . The ellipticity of the operator (see [9]) and smoothness in of (since ) imply .

Similarly, coercivity of ([7], page 69) implies infinite smoothness of in .

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgment

The research was supported by the Russian Science Foundation (Project no. 14-11-00344).

#### References

- D. L. Colton and R. Kress,
*Integral Equation Methods in Scattering Theory*, John Wiley & Sons, New York, NY, USA, 1983. View at MathSciNet - M. Costabel and E. Stephan, “A direct boundary integral equation method for transmission problems,”
*Journal of Mathematical Analysis and Applications*, vol. 106, no. 2, pp. 367–413, 1985. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - M. Costabel, “Boundary integral operators on Lipschitz domains: elementary results,”
*SIAM Journal on Mathematical Analysis*, vol. 19, no. 3, pp. 613–626, 1988. View at Publisher · View at Google Scholar · View at MathSciNet - E. P. Stephan, “Boundary integral equations for screen problems in ${\mathbb{R}}^{3}$,”
*Integral Equations and Potential Theory*, vol. 10, no. 10, pp. 236–257, 1987. View at Google Scholar - A. Kirsch and A. Lechleiter, “The operator equations of Lippmann-Schwinger type for acoustic and electromagnetic scattering problems in
*L*^{2},”*Applicable Analysis*, vol. 88, no. 6, pp. 807–830, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - M. Costabel, E. Darrigrand, and E. H. Koné, “Volume and surface integral equations for electromagnetic scattering by a dielectric body,”
*Journal of Computational and Applied Mathematics*, vol. 234, no. 6, pp. 1817–1825, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - A. S. Ilinsky and Y. G. Smirnov,
*Electromagnetic Wave Diffraction by Conducting Screens*, VPS, Utrecht, The Netherlands, 1998. - Y. G. Smirnov and A. A. Tsupak, “Investigation of electromagnetic diffraction by a dielectric body using the volume singular integral equation,”
*Computational Mathematics and Mathematical Physics*, vol. 44, no. 12, pp. 2143–2158, 2004. View at Google Scholar - D. V. Valovik and Y. G. Smirnov, “Pseudodifferential operator method in a problem of the diffraction of an electromagnetic wave on a dielectric body,”
*Differential Equations*, vol. 48, no. 4, pp. 517–523, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - M. Y. Medvedik, Y. G. Smirnov, and A. A. Tsupak, “Scalar problem of plane wave diffraction by a system of nonintersecting screens and inhomogeneous bodies,”
*Computational Mathematics and Mathematical Physics*, vol. 54, no. 8, pp. 1280–1292, 2014. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. B. Samokhin,
*Integral Equations and Iteration Methods in Electromagnetic Scattering*, VSP BV, 2001.