Abstract
In this paper, we propose a fractional generalization of the well-known Laguerre differential equation. We replace the integer derivative by the conformable fractional derivative of order $0 < \alpha<1.$ 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 solutions 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.
Highlights
In recent years, there are interests in studying fractional Sturm-Liouville eigenvalue problems
The fractional Bessel equation with applications was investigated in Okrasinski and Plociniczak [1, 2], where the fractional derivative is of the Riemann-Liouville type
Since the fractional Laguerre functions are orthogonal, they can be used as a basis of the spectral method to study fractional differential equations analytically and numerically
Summary
There are interests in studying fractional Sturm-Liouville eigenvalue problems. In Abu Hammad and Khalil [3] the authors solved the fractional Legendre equation with conformable derivative and established the orthogonality property of the fractional Legendre functions. The Fractional Laguerre Equation of Conformable Type properties of the integer derivative such as, the product rule, the quotient rule, and the chain rule, and it holds that. The rest of the paper is organized as follows: In section 2, we apply the Frobenius method together with the fractional series solution to solve the above equation and to obtain the fractional Laguerre functions. Laguerre equation with integer derivative α = 1, we obtain only one value of r = 0, which produces only one solution. (−1)mm! (n!)2(m − n)! , which is the expansion of the Laguerre polynomial Lm(x)
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