On the volume singular integro-differential equation approach for the electromagnetic diffraction problem

Abstract

The boundary value problem of diffraction electromagnetic waves on a 3-dimensional inhomogeneous dielectric body in a free space is considered. This problem is reduced to a volume singular integro-differential equation. The smoothness properties of solutions of the integro-differential equation are studied. It is proved that for smooth data the solution from will necessary be continuous down to the boundary of the body and smooth inside the body. The smoothness properties allow one to prove the equivalency between the boundary value problem and the integro-differential equation. In addition, using pseudodifferential operators calculus, an asymptotic expansion of the operator’s symbol is obtained and ellipticity and Fredholm property with zero index of the operator of the problem are proved.

Keywords:

• Maxwell’s equations
• diffraction problem
• boundary value problem
• volume singular integral equation
• pseudodifferential operator

AMS Subject Classifications:

• 35Q61
• 78A45
• 31B10
• 35S15

Notes

No potential conflict of interest was reported by the authors.

We indicate that results of Section 3 of this paper has previously been published as a part of [7]. However, we found it necessary to add them here as the theorems about ellipticity and Fredholm property and the theorem of equivalence complement each other.

Additional information

Funding

This work was supported by the Russian Science Foundation under [grant number 14-11-00344].