Some analytical properties of the Mittag-Leffler functions, e α (t) ≡ E α (−t α ), are established on some t-intervals. These are lower and upper bounds obtained in terms of simple rational functions (which are related to the Padé approximants of e α (t)), for t > 0 and 0 < α < 1. A new method, to compute such functions, solving numerically a Caputo-type fractional differential equation satisfied by them, is developed. This approach consists in an adaptive predictor-corrector method, based on the K. Diethelm's predictor-corrector algorithm, and is shown to outperform the current methods implemented by MATLAB R and by Mathematica when t is real and even possibly large.MSC 2010 : Primary 33E12; Secondary 34A08, 65L99 Key Words and Phrases: Mittag-Leffler functions, fractional ordinary differential equations, predictor-corrector methods for fractional differential equations, adaptive methods for fractional differential equations
We consider fractional relaxation and fractional oscillation equations involving Erdélyi-Kober integrals. In terms of Riemann-Liouville integrals, the equations we analyze can be understood as equations with timevarying coefficients. Replacing Riemann-Liouville integrals with Erdélyi-Kober-type integrals in certain fractional oscillation models, we obtain some more general integro-differential equations. The corresponding Cauchytype problems can be solved numerically, and, in some cases analytically, in terms of Saigo-Kilbas Mittag-Leffler functions. The numerical results are obtained by a treatment similar to that developed by K. Diethelm and N.J. Ford to solve the Bagley-Torvik equation. Novel results about the numerical approach to the fractional damped oscillator equation with time-varying coefficients are also presented.MSC 2010 : Primary 34C26; 26A33; 65L05; Secondary 326A33; 26A48; 34A08; 33E12
An attempt is made to identify the orders of the fractional derivatives in a simple anomalous diffusion model, starting from real data. We consider experimental data taken at the Columbus Air Force Base in Mississippi. Using as a model a one-dimensional fractional diffusion equation in both space and time, we fit the data by choosing several values of the fractional orders and computing the infinite-norm "errors", representing the discrepancy between the numerical solution to the model equation and the experimental data. Data were also filtered before being used, to see possible improvements. The minimal discrepancy is attained correspondingly to a fractional order in time around 0.6 and a fractional order in space near 2. These results may describe well the memory properties of the porous medium that can be observed.
A numerical method for solving fractional partial differential equations (fPDEs) of the diffusion and reaction–diffusion type, subject to Dirichlet boundary data, in three dimensions is developed. Such fPDEs may describe fluid flows through porous media better than classical diffusion equations. This is a new, fractional version of the Alternating Direction Implicit (ADI) method, where the source term is balanced, in that its effect is split in the three space directions, and it may be relevant, especially in the case of anisotropy. The method is unconditionally stable, second-order in space, and third-order in time. A strategy is devised in order to improve its speed of convergence by means of an extrapolation method that is coupled to the PageRank algorithm. Some numerical examples are given.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.