The Sommerfeld problem and inverse problem for the Helmholtz equation
Kalmenov T.S. Kabanikhin S.I. Les A.
1 February 2021De Gruyter Open Ltd
Journal of Inverse and Ill-Posed Problems
2021#29Issue 149 - 64 pp.
The study of a time-periodic solution of the multidimensional wave equation ∂2∂t2 u∼-Δx2 u∼ = f∼ (x,t),u∼(x, t) = e kitu(x), over the whole space ℝ3 leads to the condition of the Sommerfeld radiation at infinity. This is a problem that describes the motion of scattering stationary waves from a source that is in a bounded area. The inverse problem of finding this source is equivalent to reducing the Sommerfeld problem to a boundary value problem for the Helmholtz equation in a finite domain. Therefore, the Sommerfeld problem is a special inverse problem. It should be noted that in the work of Bezmenov [I. V. Bezmenov, Transfer of Sommerfeld radiation conditions to an artificial boundary of the region based on the variational principle, Sb. Math. 185 1995, 3, 3-24] approximate forms of such boundary conditions were found. In [T. S. Kalmenov and D. Suragan, Transfer of Sommerfeld radiation conditions to the boundary of a limited area, J. Comput. Math. Math. Phys. 52 2012, 6, 1063-1068], for a complex parameter λ, an explicit form of these boundary conditions was found through the boundary condition of the Helmholtz potential given by the integral in the finite domain ω: u(x,λ)= ∫ ωϵ (x-ξ,λ) ρ(ξ,λ),d ξ where ϵ (x-ξ,λ) are fundamental solutions of the Helmholtz equation, -Δx ϵ (x)-λ ϵ = δ (x), ρ (ξ,λ) is a density of the potential, λ is a complex number, and δis the Dirac delta function. These boundary conditions have the property that stationary waves coming from the region ω to ∂ ω pass ∂ ω without reflection, i.e. are transparent boundary conditions. In the present work, in the general case, in ℝn, n≥3, we have proved the problem of reducing the Sommerfeld problem to a boundary value problem in a finite domain. Under the necessary conditions for the Helmholtz potential (∗), its density ρ (ξ,λ) has also been found.
Helmholtz equation , Inverse problems , Sommerfeld problem
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Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan
Institute of Computational Mathematics and Mathematical Geophysics SB RAS, Novosibirsk, Russian Federation
Institute of Mathematics and Mathematical Modeling
Institute of Computational Mathematics and Mathematical Geophysics SB RAS
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