Abstract
We consider a boundary value problem of the stationary transport equation with the incoming boundary condition in two or three dimensional bounded convex domains. We discuss discontinuity of the solution to the boundary value problem arising from discontinuous incoming boundary data, which we call the boundary-induced discontinuity. In particular, we give two kinds of sufficient conditions on the incoming boundary data for the boundary-induced discontinuity. We propose a method to reconstruct the attenuation coefficient from jumps in boundary measurements.
Highlights
We consider the stationary transport equation:ξ · ∇xf (x, ξ) + μt(x)f (x, ξ) = μs(x) p(x, ξ, ξ′)f (x, ξ′) dσξ′ . (1) S d−1The stationary transport equation describes propagation of photons [6]
We introduce a boundary value problem to the equation (1)
Our aim in this paper is to propose a way to reconstruct the attenuation coefficient μt from the boundary data, f0 and f |Γ+, of the solution f to the boundary value problem (1)-(2) with μs and p unknown
Summary
We call a bounded function f on D a solution to the boundary value problem (1)-(2) if (i) it has the directional derivative ξ · ∇xf (x, ξ) at all (x, ξ) ∈ Ω0 × Sd−1, (ii) it satisfies the stationary transport equation (1) for all (x, ξ) ∈ Ω0 × Sd−1 and the boundary condition (2) for all (x, ξ) ∈ Γ−, (iii) f (·, ξ) is continuous along the line {x + tξ|t ∈ R} ∩ (Ω ∪ Γ−,ξ) for all (x, ξ) ∈ D, and (iv) ξ · ∇xf (·, ξ) is continuous on the open line segments {x + tξ|t ∈ (tj−1(x, ξ), tj (x, ξ))}, j = 1, . Let f be the extended solution to the boundary value problem (1)-(2) with the incoming boundary data given by (3), and let (x∗, ξ∗) ∈ disc(f ).
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
