By Mezhlum A. Sumbatyan
Equations of Mathematical Diffraction idea specializes in the comparative research and improvement of effective analytical tools for fixing equations of mathematical diffraction thought. Following an outline of a few normal homes of necessary and differential operators within the context of the linear thought of diffraction approaches, the authors supply estimates of the operator norms for numerous levels of the wave quantity edition, after which research the spectral houses of those operators. additionally they current a brand new analytical approach for developing asymptotic ideas of boundary indispensable equations in mathematical diffraction concept for the high-frequency case. truly demonstrating the shut connection among heuristic and rigorous tools in mathematical diffraction conception, this precious ebook provide you with the differential and crucial equations which can simply be utilized in useful purposes.
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Extra resources for Equations of Mathematical Diffraction Theory
Thus, the better the kernel the better qualitative properties of the equation. By contrast, equations of the first kind with regular kernels are, as a rule, unsolvable. Therefore we are faced here with an astonishing phenomenon—the worse the kernel the better qualitative properties of the equation. 4◦ . The most interesting and important for applications in diffraction theory is the case when 1 D 1+O , s → ∞. 140) L(s ) = |s | s Unfortunately, the theory discussed here states a (weak) solvability of Eq.
178) is called a hyper-singular integral . First of all, we should clarify in which sense hyper-singular integrals as given by Eq. 178) may be treated, since they exist neither as improper integrals of the first kind nor as Cauchy-type singular integrals. At least three different definitions of hyper-singular integrals are known (see Samko, 2000): 1. The integral is a derivative of the Cauchy principal value: a –a d u(ξ ) dξ =– 2 (x – ξ ) dx a –a u (ξ ) dξ . 179) 2. The integral is treated as a Hadamard principal value (see Belotserkovsky and Lifanov, 1993): a –a x–ε u(ξ ) dξ = lim (x – ξ )2 ε→ +0 a + –a x+ε u (ξ ) dξ 2u(x) .
Thus, out of two plane waves in Eq. 211), only the first one satisfies the radiation condition. 211)) coincide at the points defined by the radius vector r and the radius vector r + λn. This implies λk = 2π , or c 2π 2πc = = , where ω = 2πf . 214) λ= k ω f Here ω is the angular frequency, which is measured in rad/s, and f is called the cyclic frequency and is measured in Hz = 1/s. In diffraction problems where an incident wave falls onto an obstacle, complete mathematical formulation implies some boundary conditions on the boundary surface (3D case) or boundary line (2D case) of the obstacle.