Existence and uniqueness of solutions of stationary transport equations |
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| Authors: | Illya Karabash |
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| Affiliation: | Department of Partial Differential Equations, The Institute of Applied Mathematics and Mechanics of NAS of Ukraine, R. Luxemburg str.74, 83114 Donetsk, Ukraine |
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| Abstract: | We consider the abstract kinetic equation dψ /dx = –JLψ, x ∈ [0, τ ], in a Hilbert space H. It is supposed that J = J * = J–1, L = L * ≥ 0, ker L = 0. The following theorem is proved: if JL is similar to a self-adjoint operator, then an associated boundary problem has a unique solution. We apply this theorem to the stationary equation of Brownian motion (sgn μ)|μ |α (∂ψ /∂x) (x,μ) = (∂2ψ /∂μ2) (x,μ), 0 < x < τ, μ ∈ ℝ. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) |
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