647 research outputs found

    Numerical Results for the Generalized Mittag-Leffler Function

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    Mathematics Subject Classification: 33E12, 33FXX PACS (Physics Abstracts Classification Scheme): 02.30.Gp, 02.60.GfResults of extensive calculations for the generalized Mittag-Leffler function E0.8,0.9(z) are presented in the region −8 ≤ Re z ≤ 5 and −10 ≤ Im z ≤ 10 of the complex plane. This function is related to the eigenfunction of a fractional derivative of order α = 0.8 and type β = 0.5

    Computation of the generalized Mittag-Leffler function and its inverse in the complex plane

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    The generalized Mittag-Leffler function E α,β (z) has been studied for arbitrary complex argument z ∈ C and parameters α ∈ R + and β ∈ R. This function plays a fundamental role in the theory of fractional differential equations and numerous applications in physics. The Mittag-Leffler function interpolates smoothly between exponential and algebraic functional behaviour. A numerical algorithm for its evaluation has been developed. The algorithm is based on integral representations and exponential asymptotics. Results of extensive numerical calculations for E α,β (z) in the complex z-plane are reported here. We find that all complex zeros emerge from the point z = 1 for small α. They diverge towards −∞ + (2k − 1)πi for α → 1 − and towards −∞ + 2kπi for α → 1 + (k ∈ Z). All the complex zeros collapse pairwise onto the negative real axis for α → 2. We introduce and study also the inverse generalized Mittag-Leffler function L α,β (z) defined as the solution of the equation L α,β (E α,β (z)) = z. We determine its principal branch numerically

    Galerkin FEM for fractional order parabolic equations with initial data in H−s, 0<s≤1H^{-s},~0 < s \le 1

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    We investigate semi-discrete numerical schemes based on the standard Galerkin and lumped mass Galerkin finite element methods for an initial-boundary value problem for homogeneous fractional diffusion problems with non-smooth initial data. We assume that Ω⊂Rd\Omega\subset \mathbb{R}^d, d=1,2,3d=1,2,3 is a convex polygonal (polyhedral) domain. We theoretically justify optimal order error estimates in L2L_2- and H1H^1-norms for initial data in H−s(Ω), 0≤s≤1H^{-s}(\Omega),~0\le s \le 1. We confirm our theoretical findings with a number of numerical tests that include initial data vv being a Dirac δ\delta-function supported on a (d−1)(d-1)-dimensional manifold.Comment: 13 pages, 3 figure

    Infrared spectroscopy of diatomic molecules - a fractional calculus approach

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    The eigenvalue spectrum of the fractional quantum harmonic oscillator is calculated numerically solving the fractional Schr\"odinger equation based on the Riemann and Caputo definition of a fractional derivative. The fractional approach allows a smooth transition between vibrational and rotational type spectra, which is shown to be an appropriate tool to analyze IR spectra of diatomic molecules.Comment: revised + extended version, 9 pages, 6 figure
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