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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)
##### MSC:
 45K05 Integro-partial differential equations