Recently, several authors demonstrated the usefulness of fractional calculus operators in the derivation of particular solutions of a considerably large number of linear ordinary and partial differential equations of the second and higher orders. By means of fractional calculus techniques, we find explicit solutions of second-order linear ordinary differential equations. Recently, several authors demonstrated the usefulness of fractional calculus operators in the derivation of particular solutions of a considerably large number of linear ordinary and partial differential equations of the second and higher orders. By means of fractional calculus techniques, we find explicit solutions of second-order linear ordinary differential equations.

The fractional calculus includes concepts of integrals and derivatives of any complex or real order. The fractional calculus is as old as the usual calculus. Recently, many scientists have been studying on this field to provide the development and applicability to various areas of mathematics, physics, engineering and other sciences. The discrete fractional calculus has an important position in this field. Some ordinary diffrential equations can be solved by means of nabla (∇) discrete fractional calculus operator. In this paper, we aim to apply this operator to the radial Schrödinger equation for some physical potentials such as pseudoharmonic and Mie-type potentials.

In thermal sciences, it is possible to encounter topics such as Bessel beams, Bessel functions or Bessel equations. In this work, we also present new discrete fractional solutions of the modified Bessel differential equation by means of the r-discrete fractional calculus operator. We consider homogeneous and non-homogeneous modified Bessel differential equation. So, we acquire four new solutions of these equations in the discrete fractional forms via a newly developed method.