We solve the inverse spectral problems for the class of Sturm–Liouville operators with singular real-valued potentials from the Sobolev space W s−1 2 (0, 1), s ∈ [0, 1]. The potential is recovered from two spectra or from the spectrum and norming constants. Necessary and sufficient conditions on the spectral data to correspond to the potential in W s−1 2 (0, 1) are established.

The paper addresses nonlinear inverse Sturm–Liouville-type problems with constant delay. Since many processes in the real world possess nonlocal nature, operators delay as well other classes of are continuously finding numerous applications natural sciences and engineering. However, spite a large number works devoted to for delay, existing results do not give comprehensive picture all values pa...

We consider the Sturm–Liouville equation on a finite interval with real-valued integrable potential and propose method for solving following general inverse problem. recover from given set of output boundary values solution satisfying some known initial conditions spectral parameter. Special cases this problem include recovery Weyl function, two-spectra problem, as well plane wave that interact...

We discuss direct and inverse spectral theory for a Sturm–Liouville type problem with a quadratic dependence on the eigenvalue parameter, −f ′′ + 1 4 f = z ωf + zυf, which arises as the isospectral problem for the conservative Camassa–Holm flow. In order to be able to treat rather irregular coefficients (that is, when ω is a real-valued Borel measure on R and υ is a non-negative Borel measure o...

p−1 p p−1 . This equation is a special case of a general half-linear second order differential equation (r(t)Φ(x′))′ + c(t)Φ(x) = 0, (2) where Φ(x) := |x| sgn x, p > 1, and r, c are continuous functions, r(t) > 0 (in the studied equation (1) we have r(t) ≡ 1). Let us recall that similarly as in the linear case, which is a special case of (2) for p = 2 and equation (2) then reduces to the linear...

Abstract. We introduce a variational algorithm, which solves the classical inverse Sturm-Liouville problem when two spectra are given. In contrast to other approaches, it recovers the potential as well as the boundary conditions without a priori knowledge of the mean of the potential. Numerical examples show that the algorithm works quite reliable, even in the presence of noise. A proof of the ...

Abstract. We present a variational algorithm for solving the classical inverse Sturm-Liouville problem in one dimension when two spectra are given. All critical points of the least squares functional are at global minima, which justifies minimization by a (conjugate) gradient descent algorithm. Numerical examples show that the resulting algorithm works quite reliable without tuning for particul...

In this study, we obtained a system of eigenfunctions and eigenvalues for the mixed homogeneous Sturm-Liouville problem second-order differential equation containing fractional derivative operator. The differentiation operator was considered according to two definitions: Gerasimov-Caputo Riemann-Liouville-Visualizations eigenfunctions, biorthogonal system, distribution on real axis were present...