In this study we consider the Kaup–Boussinesq system with a self-consistent source. We show that source can be integrated by method of inverse scattering theory. For solving problem under consideration, use direct and Sturm–Liouville equation an energy-dependent potential. The time evolution data for potentials associated solution is determined. obtained equalities completely determine any $t$,...

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...

We consider the Sturm–Liouville operator perturbed by a convolution integral on finite interval with Dirichlet boundary conditions and discontinuity in middle of interval. study inverse problem recovering term from spectrum. The is reduced to solving so-called main nonlinear equation singularity. To derive this equation, we do detailed analysis kernels transformation operators for integro-diffe...

  Abstract.   The Sturm-Liouville boundary value problem of the multi-order fractional differential equation  is studied. Results on the existence of solutions are established. The analysis relies on a weighted function space and a fixed point theorem. An example is given to illustrate the efficiency of the main theorems.

We study the Al-Salam–Chihara polynomials when q > 1. Several solutions of the associated moment problem are found, and the orthogonality relations lead to explicit evaluations of several integrals. The polynomials are shown to have raising and lowering operators and a second order operator equation of Sturm-Liouville type whose eigenvalues are found explicitly. We also derive new measures with...