Numerical treatment of fractional differential equations

Abstract

In Chapter 1, we gave a partial contribution to the Mainardi's conjecture, concerning only small intervals of the variable t. The method we proposed to evaluate numerically the M-L function e_(t) _ E_(􀀀t_), is based on the integration of a simple fODE satis_ed by it. We did it using the predictorcorrector algorithm of K. Diethelm, and showed that it always wins, in terms of CPU time and RAM, over the code implemented by Mathematica code, for all values of _ 2 (0; 1) and for arbitrary times. It also wins over the MATLAB code with some limitations on _ and t. An adaptive predictorcorrector algorithm we then implemented, however, always outperformed even the MATLAB code, for all _ and t. No comparison can be made with the evaluation of the more general two-parameters M-L functions as well as with that of the one-parameter M-L function on the complex domain, that we did not consider. In Chapter 1, a few models describing fractional relaxation as well as fractional oscillations have been studied. These are based on simple fractional fODEs, some being related to the so-called _Erdelyi-Kober operators. In Chapter 2, we compared the numerical results, obtained modeling onedimensional anomalous di_usion by fPDEs, with some experimental (hence realistic) data, and showed that the numerical solution of the aforementioned fPDE _ts the laboratory results better than using classical PDEs (with integer order derivatives). More precisely, we found that, in the cases considered, the results match within a small error, both in L2 and in L1 norm, when the space fractional order is slightly below 2 (say, about 1:9, but the time fractional exponent seems to be optimal choosing a value around 0:6 􀀀 0:7. This shows that indeed several porous media exhibit memory e_ects and the departure from the classical description is evident. In Chapter 3, a weighted and shifted Gr unwald-Letnikov di_erence (WSGD) operator is used to approximate RL fractional di_usion operators. It is shown that indeed third-order accuracy in time can be achieved solving numerically two-dimensional fPDEs by ADI-like methods. A new technique, designed to accelerate the algorithm, which is competitive with respect to the methods existing to date in the literature [3], has also been developed. While the present method seems to outperform all the other existing algorithms, using very dense grids to attain low errors may require, however, as one may expect, a considerable computational time. In Chapter 4 we introduced a new well \balanced", fractional version of the ADI method, to solve numerically a number of 3D di_usion as well as reaction-di_usion problems for fPDEs. These are important in a variety of problems arising from porous media modeling. We proved that such a method is unconditionally stable for every fractional order of the space derivatives, second-order accurate in space, and third-order accurate in time. The speed of convergence has been improved adopting an extrapolation technique, coupled with the optimization method used by the PageRank algorithm. Future directions can easily be envisaged, since nowadays fPDEs seem to be relevant in many other _elds, such as Economy and Finance, Biology and Demography, are being involved beside the more traditional of Viscoelasticity and Seismology. There is an increasing demand for numerical methods to tackle complex problems in several dimensions. Researchers, especially mathematicians, should resist the temptation of \fractionalizing" every classical model equation, checking _rst whether this kind of generalization ha sound (e.g. physical) bases which motivates this choice. It may be however useful to explore what could be extracted, which further or better explanation could be inferred form such approach. Con_ning to the models considered in Chapter 3 and Chapter 4, it would be interesting to study the e_ects of anisotropy due to di_erent fractional orders a_ecting di_erent space directions. Another sensible topic is the identi_cation of the fractional orders themselves (inverse problems), in a given fractional di_erential equation. To date, very little can be found in the literature [1, 2]. Such identi_cation was attempted, over the years, in a few laboratory experiments, hence basing on real measurements, e.g., for anomalous di_usion in several kinds of porous media [4, 5].