A new algorithm is proposed for solving the inverse Sturm–Liouville problem of reconstructing a symmetric potential from eigenvalues. It uses Numerov’s method instead of the second order method of the related algorithm of Fabiano, Knobel and Lowe. An extension by Andrew and Paine of the asymptotic correction technique of Paine, de Hoog and Anderssen is the key to the success of the new algorith...

In this lecture we abstract the eigenvalue problems that we have found so useful thus far for solving the PDEs to a general class of boundary value problems that share a common set of properties. The so-called Sturm-Liouville Problems define a class of eigenvalue problems, which include many of the previous problems as special cases. The S − L Problem helps to identify those assumptions that ar...

Uniqueness of and numerical techniques for the inverse Sturm-Liouville problem with eigenparameter dependent boundary conditions will be discussed. We will use a Gel’fand-Levitan technique to show that the potential q in u00 þ qu 1⁄4 u, 0 < x < 1 uð0Þ 1⁄4 0, ða þ bÞuð1Þ 1⁄4 ðc þ d Þu0ð1Þ can be uniquely determined using spectral data. In the presence of finite spectral data, q can be reconstruc...

Abstract In this paper, we are concerned with the eigenvalue gap and ratio of Dirichlet conformable fractional Sturm–Liouville problems. We show that kind differential equation satisfies property by Prüfer substitution. That is, n th eigenfunction has $n-1$ n − 1 zero in $( 0...

We prove a necessary optimality condition for isoperimetric problems on time scales in the space of delta-differentiable functions with rdcontinuous derivatives. The results are then applied to Sturm-Liouville eigenvalue problems on time scales.

This paper concerns the nonlinear Sturm-Liouville problem −u′′(t) + f(u(t)) = λu(t), u(t) > 0, t ∈ I := (0, 1), u(0) = u(1) = 0, where λ is a positive parameter. We try to determine the nonlinear term f(u) by means of the global behavior of the bifurcation branch of the positive solutions in R+ × L2(I).

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Simple and analytically tractable expressions for functional determinants are known to exist for many cases of interest. We extend the range of situations for which these hold to cover systems of self-adjoint operators of the Sturm-Liouville type with arbitrary linear boundary conditions. The results hold whether or not the operators have negative eigenvalues. The physically important case of f...