Inverse nodal problems for Sturm - Liouville equations with eigenparameter dependent boundary conditions

Abstract

Recent results of Hald and McLaughlin concerning the inverse problem for the regular Sturm-Liouville problem on a finite interval are extended to the case in which the boundary conditions are eigenparameter dependent. Specifically, we show that the potential and the 'asymptotic' boundary conditions in such a problem are uniquely determined by a dense set of nodal points of eigenfunctions. We also simplify the proofs given by Hald and McLaughlin.

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

“…This study includes discontinuity conditions at the middle of interval. Inverse nodal problem for SturmLiouville operator with boundary conditions dependent on the spectral parameter were investigated in [4], [23] and [18]. Additionally, the studies [5] and [6] include inverse nodal problems for di¤erential pencils.…”

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

“…Firstly, the idea of this area appeared in a letter by Leibniz to L' Hospital in 17 th century, Podlubny (1999) . N. Sneddon, S. G. Samko, T. J. Osler, E. R. Love, and many others Boumenir and Tuan, (2010a and b), Chechkin et al (2003), Freiling and Yurko (2001) and Gorenflo et al (2002). Fractional diffusion equations have been investigated in a lot of different physical situations.…”

“…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.…”

Reconstruction of the Sturm-Liouville operators on a graph with δ′ s couplings

“…From the physical point of view this corresponds to finding, e.g., the density of a string or a beam from the zero-amplitude positions of their eigenvibrations. Recently, some authors have reconstructed the potential function for generalizations of the Sturm-Liouville problem from the nodal points (for example, refer to [3,5,7,10,11,14,19,24,25,26,30,33,34,35,37,41,42]). …”