China
Africa
Explore chapters and articles related to this topic
Diffraction by a bounded planar screen
Published in A.S. Ilyinsky, Yu.G. Smirnov, Yu.V. Shestopalov, Electromagnetic Wave Diffraction by Conducting Screens, 2022
A.S. Ilyinsky, Yu.G. Smirnov, Yu.V. Shestopalov
The proof that T and L are Fredholm operators with a zero index (in the corresponding spaces) constitutes a central part of this section. We recall that a bounded operator F with a closed domain of values is called the Fredholm operator, if dim ker F<∞ and dim coker F<∞ the difference ind F = dim ker F —dim coker F is the index of operator F. In order to prove that F is a Fredholm operator with a zero index, it is sufficient to show that F = S + K, where S is a continuously invertible operator and K is a compact operator [25, 26]). We will prove that T and L are Fredholm operators using an important property of form (2.34), which can be directly verified: t(u, v) = 0 when u≠Wi and v≠Wj for i≠j,
Fredholm Theory
Published in Eberhard Malkowsky, Vladimir Rakočević, Advanced Functional Analysis, 2019
Eberhard Malkowsky, Vladimir Rakočević
Fredholm operators, the basic objects in Fredholm operator theory, are generalizations of the operators of the form A = I − K, where K∈K(X) and X is a Banach space. They are very important for the spectral theory of operators. In this chapter we present some elements from Fredholm theory. We refer the interested readers to the textbooks and monographs [2,3,15,38,101,112,142] for further reading.
Fredholm Integral Equation in Electrochemistry
Published in Mihai V. Putz, New Frontiers in Nanochemistry, 2020
Mirela I. Iorga, Mihai V. Putz
Fredholm integral equation represents in mathematics an integral equation whose solutions have to give rise to Fredholm theory, Fredholm nuclei study, and Fredholm operators. Fredholm integral equation was the basis of Fredholm’s theorems, one of them being known as Fredholm alternative. The applicability of Fredholm integral equation was demonstrated in several domains, among them one of great importance is electrochemistry because the solutions of Fredholm integral equations facilitate the determination of process conditions, parameters, and so on.
Non-iterative two-step method for solving scalar inverse 3D diffraction problem
Published in Inverse Problems in Science and Engineering, 2020
M. Yu. Medvedik, Yu. G. Smirnov, A. A. Tsupak
In Section 2, we rigorously formulate the forward problem of diffraction and give main theoretical results of its investigation. Note that some results are similar to the ones described in [15]. The major difference from [15] is that we consider the quasiclassical setting of the boundary value problem (BVP) for the Helmholtz equation, assuming that a solution should satisfy additional smoothness conditions. Then, we write the integral Lippmann–Schwinger equation (LSIE) with respect to the unknown total field in the inhomogeneity region Q. The operator of the equation is well studied [15] and is known to be a Fredholm operator of index zero. Further, we show that any solution to LSIE with a smooth right-hand side represents in fact a quasiclassical solution to the original BVP. Thus, we obtain that the BVP for the Helmholtz equation is equivalent to the system of integral equations used for recovering the refractive index in Section 3 of the article.
The two-step method for determining a piecewise-continuous refractive index of a 2D scatterer by near field measurements
Published in Inverse Problems in Science and Engineering, 2020
M. Yu. Medvedik, Yu. G. Smirnov, A. A. Tsupak
Since for any we have then is a compact operator in Consequently, is a Fredholm operator with index zero.