The method of straight lines for solving the non-stationary transport equation

Abstract

WE prove in this paper an existence theorem for the solution of the transport integro-differential equation for an infinite plane strip G: {− a ⩽ z ⩽ a; − 1 ⩽ μ ⩽ + 1, t ⩾ 0}: (1) 1 v ∂I(z,μ,t ∂t + μ ∂I(z,μ,t) ∂z + σI (z,μ, t) = λ ∝ −1 1 K(μ,μ′)I(z,μ′,)dμ′ (see e.g. [1], satisfying the initial condition I( z, μ, 0) = g( z, μ) and the boundary conditions = 0, μ > 0, I( a, μ, t) = 0, μ < 0 (cf. Theorem 4), and also show that the approximate solution of this problem, obtained by the method of straight lines, is convergent to the exact solution; further, some error estimates are obtained, according to the functional properties of the kernel (cf. Theorems 1–3). Notice that similar results can be obtained by the same method (see the proofs of Theorems 1–4) for more complex non-stationary transport equations e.g. for the case of a homogeneous sphere. The existence of a solution for certain problems of transport theory is proved in [2, 3].