The Fractional Laguerre Equation: Series Solutions and Fractional Laguerre Functions

Abstract

In this paper, we propose a fractional generalization of the well-known Laguerre differential equation. We replace the integer derivative by the conformable derivative of order 0 < α < 1. We then apply the Frobenius method with the fractional power series expansion to obtain two linearly independent solutions of the problem. For certain eigenvalues, the infinite series solution truncate to obtain the singular and non-singular fractional Laguerre functions. We obtain the fractional Laguerre functions in closed forms, and establish their orthogonality result. The applicability of the new fractional Laguerre functions is illustrated.

Complete and orthonormal sets of exponential-type orbitals with non-integer quantum numbers

“…A power series solution for equation ( 60) was devised in [72]. The conformable local derivative requires that the power series expansion be clearly separated, between integer and fractional parts as qϵ.…”

Solution of Fractional Partial Differential Equations Using Fractional Power Series Method

“…In order to validate the high degree of efficiency and precision of the projected FPS approach for unraveling fractional-order systems, numerical forms and instances are pragmatic. e reader can discover a sketch and applications for this technique in [42]. Computations were accomplished by using MATLAB.…”

“…In nonfractional regular linear ODEs, with different coefficients, the series expansion method is considered to help clarify the problem. Examples of power series can be seen in [34][35][36][37][38][39][40][41][42][43][44][45][46][47][48]. e description of this theory is also related to the use of the residual power series method [49] to solve spacetime fractional PDEs.…”