On the determination of point sources by boundary observations: uniqueness, stability and reconstruction
- Bruckner, Gottfried
- Yamamoto, Masahiro
2010 Mathematics Subject Classification
- 35R30 35L20 73D50
- Determination of point sources, vibration of a string, boundary measurements, uniqueness, stability, reconstruction
We consider the problem
u''(x,t) = uxx(x,t) + λ(t) ΣNk=1αkδ(x-ξk), 0 < x < 1,0 < t < T
u(x,0) = u'(x,0) = 0, 0 < x < 1
u(0,t) = u(1,t) = 0, 0 < t < T,
where u'(x,t) = ∂u ⁄ ∂t, (x,t), u'' (x,t) = ∂2u ⁄ ∂t2 (x,t), and λ ∈ C1[0,T], αk ≠ 0, ∈ ℝ, ξk ∈ (0,1), and δ(·-ξk) is Dirac's delta function at ξk, 1 ≤ k ≤ n. Our task consists in the determination of N, αk, ξk, 1 ≤ k ≤ N from the boundary observation ∂u ⁄ ∂x (0,t), 0 < t < T, where λ and T > 0 are given. We prove the uniqueness, give a stability estimate and provide a scheme for reconstructing α1, α2, ξ1, ξ2 from ∂u ⁄ ∂x (0,t), 0 < t < T in the case N = 2.
- Inverse Problems 16, (2000), pp. 723-748.