The paper "Algorithms for the fractional calculus: A selection of numerical methods" by Diethelm et al (2005) in Computer methods in applied mechanics and engineering, vol 194, pp 743-773 gives a very good review. In the paper's own words, it gives the newcomer "the necessary tools required to work with fractional models in an efficient way" --- this paper would be a good place to start...good luck

Hi Muhammas and hi everyone,

I suggest you to read the paper "Fractional differential equations solved by using Mellin transform" by myself anf prof. Mario Di Paola. You can find it at the following wesites:

http://arxiv.org/pdf/1402.5949.pdf

http://dx.doi.org/10.1016/j.cnsns.2013.11.022

It is about a very powerful and versatile technique to solve fractional differential equations with constant coefficients by using the mellin transform. Very breafly, what we do is to transform the equation in the complex (mellin) domain, solve for a discretization of the transformed unkown function, and recover the solution in the time domain (or in whatever other physical space) by applying the discretized version of the inverse Mellin transform.

I hope to be helpful

Salvatore

*Numerical approaches to system of fractional partial differential equations by H. F.Ahmed, Mohamed S. M.Bahgat, MofidaZaki

* Numerical Methods for Fractional Order Singular Partial Differential Equations with Variable Coefficients by Asma Ali Elbeleze, Adem Kılıçman, Bachok M. Taib

* Algorithms for the Fractional Calculus: A selection of Numerical Methods" by K.Diethelm, N.J.Ford, A.D. Freed, Yu. Luchko , Computer Methods in Applied Mechanics and Engineering, 194(2005), 743-773.

Please see the following papers

1- On fractional backward differential formulas for fractional delay differential equations with periodic and anti-periodic conditions

MS Heris, M Javidi - Applied Numerical Mathematics, 2017

2-On FBDF5 Method for Delay Differential Equations of Fractional Order with Periodic and Anti-Periodic Conditions

MS Heris, M Javidi - Mediterranean Journal of Mathematics, 2017

See these papers based on fractional differential equations.