Remarks on a New Inverse Nodal Problem

Abstract

In a recent paper, X. F. Yang proved a uniqueness theorem on inverse nodal problems that links to inverse spectral theory, on one hand, and reduces the redundancy of the classical inverse nodal problems, on the other hand. In this note we improve Yang's theorem by weakening its conditions and simplifying its proof. We also discuss variants of Yang's theorem. ᮊ

On a uniqueness theorem of Sturm–Liouville equations with boundary conditions polynomially dependent on the spectral parameter

“…For the case R 00 (λ) = 1, R 01 (λ) = -h in (1.2) and R 10 (λ) = 1, R 11 (λ) = H in (1.3), the operator U(q, U 0 , U 1 ) turns to a classical Sturm-Liouville problem L(q, h, H). Inverse spectral problems and inverse nodal problems of L(q, h, H) have been well studied, the readers can refer to [2,[10][11][12][13][14][15][16][17][18][19][20][21] and the references therein.…”

Inverse nodal problem for a Sturm-Liouville operator with discontinuous coefficient

“…From then on, their results have been generalized to various problems. Inverse nodal problems for Sturm-Liouville operators without discontinuities have been studied in the several papers ( [8], [10], [12], [13], [14], [19], [21], [22] and [24]). The …rst result on inverse nodal problems for the Sturm-Liouville operators with a discontinuity condition was obtained by Shieh and Yurko [20].…”

<b>The Inverse Nodal problem for the fractional diffusion equation

“…Hald and McLaughlin (1989) and Browne and Sleeman (1996) proved that one can use the nodal points to determine the potential function of regular Sturm-Liouville problem. In the last years, the inverse nodal problem and fractional calculus for Sturm Liouville problem has been studied by several authors Browne and Sleeman (1996), Yang (1997), Cheng et al (2000), McLaughlin (1988), Bas (2013), Koyunbakan and Panakhov (2007), Gasymov and Guseinov (1981). Tuan (2011) proved that by taking suitable initial distributions only finitely many measurements on the boundary were required to recover uniquely the diffusion coefficient of one dimensional fractional diffusion equation.…”