Tham khảo tài liệu 'electromagnetic waves part 5', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 130 Electromagnetic Waves Following the idea used for the analysis of diffraction by a strip we represent the scattered field using the fractional Green s function c x y J f 1 x Ga x -x y dx 31 0 where f1 x is the unknown function Ga is the fractional Green s function 2 . After substituting the representation 31 into fractional boundary conditions 30 we get the equation -i . . . . rnf . . 1 r 2a Ỉ 1- Í IT 1 I V 2 I . 2 I 3 r _ 1 I .a I i i ix tx lim Dky J J x H0 I kyj x x y I dx lim UkyEz x y x 0. 32 4 y - 4 V J y- The Fourier transform of f1 a x is defined as F1-a q f1- ỉ e- i dỉ f1 a x e-ikqxdx - 0 where f1-a ỉ for ỉ 0 and 0 for ỉ 0 . Then the scattered field will be expressed via the Fourier transform F1-a q as E x y ina 2 r 7. -i 4 J F 1- q eik xq y 1-q 1 -q2 a-1 2dq 33 Using the Fourier transform the equation 32 is reduced to the DIE with respect to F1 q F1-a q eikỉq 1 q2 -1 2 dq 4 e 2 1-a sin 0e-ikỉcos0 ỉ 0 - 34 J F 1- q eikỉqdq 0 ỉ 0. .- The kernels in integrals 34 are similar to the ones in DIE 17 obtained for a strip if the constant dL is equal to 1 L 0 in the case of a half-plane . For the limit cases of the fractional order a 0 and a 1 these equations are reduced to well known integral equations used for the PEC and PMC half-planes Veliev 1999 respectively. In this paper the method to solve DIE 5 is proposed for arbitrary values of a e 0 1 . DIE allows an analytical solution in the special case of a in the same manner as for a strip with fractional boundary conditions. Indeed for a we obtain the solution for any value of k as F0 5 q 2 sin1 2 0 e 4 ks q cos ớ Fractional Operators Approach and Fractional Boundary Conditions 131 f05 x 2sin1 2 ue - e s . The scattered field can be found in the following form Es x y _Le i a 2ei 4 sina 1 2 0eik -cosỡx lyn a for y 0 y 0 . In the general case of 0 a 1 the equations 34 can be reduced to SLAE. To do this we represent the unknown function f1 a a as a series in terms of .
