On the numerical solution of the eigenvalue problem in fractional quantum mechanics

Fractional Laplace Operator and Meijer G-function

“…Before we define the Meijer G-function, we discuss the role of the above conditions. Condition S is needed because it separates the poles of Γ(b j + s) from the poles of Γ(1 − a j − s) in the numerator in (13), thus the function G mn pq (a; b|s) is analytic in s in the strip −b < Re(s) < 1 − a. By Stirling's asymptotic formula for the gamma function, for λ ∈ R, …”

“…, n, so that no pole of Γ(b i + s) coincides with a pole of Γ(1 − a j − s); otherwise, the Meijer G-function is not defined. We will also always assume that p + q ≤ 2m + 2n, so that there are at least as many gamma functions in the numerator of (13) as there are in the denominator. We also introduce the following five conditions on parameters m, n, p, q, a and b that will be required for some statements:…”

A matrix method for fractional Sturm-Liouville problems on bounded domain

“…With reference to the matrix methods previously mentioned, such behavior of y determines an order reduction in the approximation of its fractional Laplacian and consequently in the resulting numerical eigenvalues. Alternative matrix methods are those proposed recently in [5,10,15]. In particular, the method in [5] is based on finite element approximations and it can be applied to problems in a generic dimension d ≥ 1, the approach considered in [10] is that of using suitable quadratures for the approximation of the integral in (3) and, finally, in [15] a Control Volume Function approximation with Radial Basis Function interpolation is proposed.…”

“…Alternative matrix methods are those proposed recently in [5,10,15]. In particular, the method in [5] is based on finite element approximations and it can be applied to problems in a generic dimension d ≥ 1, the approach considered in [10] is that of using suitable quadratures for the approximation of the integral in (3) and, finally, in [15] a Control Volume Function approximation with Radial Basis Function interpolation is proposed. All these schemes, however, appear to be of the first order, namely the error in the approximation of the eigenvalues decreases like N −1 where N is the matrix size.…”

Numerical Simulation of One-Dimensional Fractional Nonsteady Heat Transfer Model Based on the Second Kind Chebyshev Wavelet

“…Many papers on fractional calculus have been published for the real-world applications in science and engineering such as viscoelasticity [1], bioengineering [2], biology [3], and more can be found in [4,5]. Moreover fractional partial differential equations also are widely used in the areas of signal processing [6], mechanics [7], econometrics [8], fluid dynamics [9], and electromagnetics [10]. As the analytical solutions of fractional partial differential equations are not easy to derive, the scholars are committed to obtain their numerical solutions of these equations.…”