In this paper, we discuss the inverse spectral problem for Sturm–Liouville operators with boundary conditions linearly dependent on the spectral parameter and a finite number of interior discontinuities and show that if is given a priori on the interval for some , then the potential on the whole interval can be uniquely determined either by parts of a finite number of spectra, or by a finite number of subsets of pairs of eigenvalues and the norming constants of the corresponding eigenvalues. We still establish several uniqueness theorems for Sturm–Liouville operators with Robin boundary conditions and interior discontinuous conditions from the above two spectral data.

Inverse nodal problems for Sturm–Liouville equations with a constant delay are studied. The authors show that the potential up to its mean value on the whole interval can be uniquely determined by two twin-dense nodal subsets and present a constructive procedure for the solution of the above inverse nodal problem. Besides, an example shows that one twin-dense nodal subset can also reconstruct the potential up to its mean value on the whole interval for some cases, and a numerical example is presented.

In this study, integro-differential equation under separated boundary conditions is considered. In fact, we study the inverse nodal problem which includes differential part of Sturm-Liouville equation and Volterra type integral to obtain approximative solutions by Legendre polynomials. And, we use Legendre wavelets method to solve these type problems. In addition, we present the error bound of $q\left(x\right)$ for continuously derivatives with its approximative solution. Once and for all, strength of the method can be seen in a few examples.