Inverse spectral theory using nodal points as data—A uniqueness result

<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.…”

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

“…In 1988, the inverse nodal problem was posed and solved for Sturm-Liouville problems by J. R. McLaughlin [30], who showed that knowledge of a dense subset of nodal points of the eigenfunctions alone can determine the potential function of the Sturm-Liouville problem up to a constant. Some numerical schemes were provided by O. H. Hald and J. R. McLaughlin for the reconstruction of the potential [14].…”

“…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]). …”

Inverse Nodal Problem for a Class of Nonlocal Sturm‐liouville Operator

“…Such problems play an important role in mathematics and have many applications in natural sciences (see, for example, [19,21,23,32,33]). In 1988, the inverse nodal problem was posed and solved for Sturm-Liouville problems by J. R. McLaughlin [22], who showed that knowledge of a dense subset of nodal points of the eigenfunctions is sufficient to determine the potential function of the Sturm-Liouville problem up to a constant. Some numerical schemes were provided by O. H. Hald and J. R. McLaughlin [12] for the reconstruction of the potential.…”

“…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 [3,5,7,8,10,12,15,16,17,18,22,27,28,29,31,34]). When it comes to Sturm-Liouville problems with integral conditions, little has been done.…”