Space–time spectral collocation method for Klein–Gordon equation

By using the Legendre–Laguerre collocation method, we can construct a spectral collocation scheme to solve the Klein–Gordon equation on the half-line. The Laguerre function collocation method (based on the Lagrange interpolation) in space and the Legendre–Gauss–Lobatto collocation method in time are used. A Newton iterative algorithm is provided. The numerical results demonstrate the high efficiency and accuracy of suggested algorithms.

Factorization method for exact solution of the non-central modified Killingbeck potential plus a ring-shaped-like potential

In this paper, we study the Schrödinger equation with non-central modified Killingbeck potential plus a ring-shaped-like potential problem, which is not spherically symmetric. The factorization method is used to solve the hypergeometric equation types which lead to solutions with the associate Laguerre function for the radial part and Jacobi polynomial for the polar part. We introduce the raising and lowering operators to calculate the energies eigenvalues, which show that the lack of spherical symmetry removes the degeneracy of second quantum number m which is completely expected. These obtained energies are better to explain the superposition of the energy levels of the atoms in the crystalline structure of molecules.