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Determination of the transfer equation coefficient for the energy and angular properties of external radiation. (English. Russian original) Zbl 0795.45011
Sov. Phys., Dokl. 37, No. 11, 540-541 (1992); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 327, No. 2, 205-207 (1992).
Consider a bounded convex set \(G \subset R^ 3\) with smooth boundary of class \(C^ 2\), \(G=\bigcup^ p_{i=1} G_ i\), \(G_ i\) with piecewise- smooth boundaries for \(1 \leq i \leq p\) and \(G^ 0_ i \cap G^ 0_ j=\emptyset\), \(i \neq j\). Let \(\Omega=\{s;\| s \|=1\}\) be the unit sphere in \(R^ 3\). Denote by \(E\) the energy, \(0<E_ 1<E_ 2<E<+\infty\), \(\varphi=\varphi (x,s,E)\) the flux density at a point \(x \in G\) of particles moving in direction \(s \in \Omega\) with energy \(E,F=F(x,s,E)\) the intensity of interior sources, \(a=a(x,E)\) the attenuator factor and \(b=b (x,s,s',E,E')\) the scattering indicatrix.
Suppose the condition of generalized concavity be valid: if \(x \in G\) and \(s \in \Omega\), the line \(K_{x,s}=\{x+ts;t \in R\}\) intersects the boundaries of \(G_ i\), \(1 \leq i \leq p\), at a finite number of points. Let \(\sigma (x,s)\) be the distance from the point \(x \in G\) to the boundary of \(G\) in direction \(s\). Define \({\mathcal T}^ -=\{ (x,s) \in G \times \Omega\); \(\exists x_ 0 \in G,\;x=x_ 0-\sigma (x_ 0,-s)s\}\) and \({\mathcal T}^ + = \{(x,s) \in G \times \Omega\); \(\exists x_ 0 \in G,\;x=x_ 0+ \sigma (x_ 0,s)s\}\). For known \(h=h(x,s,E)\) and \(H=H(x,s,E)\), consider the problem \[ \langle s,\text{grad}_ x \varphi \rangle + a \varphi = (4\pi)^{-1} \int^{E_ 2}_{E_ 1} \int_ \Omega b \varphi ds' dE'+ F \text{ on } G \times \Omega \times [E_ 1,E_ 2], \tag{1} \] (2) \(\quad \varphi=h\) on \({\mathcal T}^ -\times [E_ 1,E_ 2]\),    (3) \(\quad \varphi=H\) on \({\mathcal T}^ +\times [E_ 1,E_ 2]\).
Suppose that the functions \(a,b,F\) satisfy some assumptions on boundedness and continuity, that there exist partial derivatives \(h_ E, H_ E\) and \(\eta \in L^ 1 [E_ 1,E_ 2]\) such that \(| h_ E (x,s,E) | \leq \eta (E)\) and there exists \(E_ 0 \in [E_ 1,E_ 2]\) such that \(\lim_{\varepsilon \to E_ 0} | h_ E (x,s,E) | =+ \infty\). Then for all \((x,s) \in {\mathcal T}^ +\) we have \[ \int^{\sigma(x,-s)}_ 0 a(x-\tau,s,E_ 0) d \tau = \ln \lim_{E \to E_ 0} h_ E \bigl( x-\sigma (x,-s)s,s,E \bigr) / H_ E(x,s,E). \tag{4} \] The unique determination of the coefficient \(a(x,E_ 0)\) from (1)–(4) is possible using Radon’s inverse transformation.
Reviewer: D.M.Bors (Iaşi)
45K05 Integro-partial differential equations