Inverse problems: Dense nodal subset on an interior subinterval

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

“…The theorem leads to the same conclusion for classical Sturm-Liouville equation when the coefficient polynomials R ij (λ) are all of degree 0 (refer to [25]), but the translation effect on boundary conditions only appears when one of R ij (λ) is a non-trivial polynomial.…”

Inverse Nodal Problem for Dirac Equations with Boundary Conditions Polynomially Dependent on the Spectral Parameter

“…The inverse nodal problem for the Dirac operator L with the known dense nodes situated only on a part of the interval is also interesting. However, it is outside the scope of this paper, the interested reader may consult [3,12,24] for recent developments. We note that the obtained results are natural generalizations of the well-known results on inverse nodal problems for the Dirac operator which were studied in [27,28].…”

“…Such problems play an important role in mathematics and have many applications in natural sciences and engineering (see [1,2,15,[20][21][22]29] and the references therein). The inverse nodal problem, first posed and solved by McLaughlin [13,23], is the problem of constructing operators from given nodes (zeros) of their eigenfunctions (refer to [3][4][5]12,14,17,24,[26][27][28]). 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.…”

Reconstruction of the Volterra-type integro-differential operator from nodal points

“…Yang provided an algorithm to determine the coefficients of the Sturm-Liouville problem by using the given nodal points in [3]. Inverse nodal problems for different types of operators have been extensively well studied in several papers (see [4][5][6][7][8][9][10][11][12][13][14] and [15]). …”