The Use of the Fractional Derivatives Approach to Solve the Problem of Diffraction of a Cylindrical Wave on an Impedance Strip

Abstract

Abstract-Earlier we considered the use of the apparatus of fractional derivatives to solve the twodimensional problem of diffraction of a plane wave on an impedance strip. We introduced the concept of a "fractional strip". A "fractional strip" is understood as a strip on the surface, which is subject to fractional boundary conditions (FBC). The problem under consideration on the basis of various methods has been studied quite well. As a rule, this problem is studied on the basis of numerical methods. The proposed approach, as will be shown below, makes it possible to obtain an analytical solution of the problem for values of fractional order ν = 0.5 and for fractional values of the interval ν ∈ [0, 1], the general solution will be investigated numerically.

Plane wave diffraction by strip with an integral boundary condition

“…After defining the fractional derivative, it is necessary to define the FBC [7][8][9][10]. U (x, y) is the function subjected to the FBC at y = d , which is the boundary of the strip in a two-dimensional case.…”

“…For fractional order, ν is equal to 0; it corresponds to a PEC surface, whereas for fractional order, ν is equal to 1 and the surface corresponds to a PMC surface. Therefore, the FBC is a more general boundary condition covering both Dirichlet and Neumann type boundary conditions [7][8][9][10].…”

“…By multiplying the integral equation in (10) by e −ixϵβ and integrating with respect to x from -a to a, IE (11) is obtained. For any arbitrary fractional order ν , IE (11) needs to be solved in order to find the Fourier transform of the fractional current density:…”

“…In order to find the scattered electric field, the total electric field needs to satisfy the boundary condition on the surface of the strip [7][8][9][10]. Here, the boundary condition is the fractional boundary condition and the mathematical expression is given as:…”

“…Using the properties of the fractional derivative approach, the intermediate stages of fields or sources between two canonical states can be described [1][2][3][4][5]. Since then, several studies have been performed on scattering problems [6][7][8][9][10]. The integral boundary condition, which corresponds to an intermediate boundary condition between Dirichlet and Neumann boundary conditions, is used in order to explain the scattering properties of different geometries.…”