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Mittag-Leffler function

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17 Oct 2005 (Updated )

Calculates the Mittag-Leffler function with desired accuracy.

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Description

This is a MATLAB routine for evaluating the Mittag-Leffler function with two parameters (sometimes also called generalized exponential function).

The Mittag-Leffler function with two parameters plays an important role and appears frequently in solutions of fractional differential equations (i.e. differential equations containing fractional derivatives).

USAGE:
MLF(alpha,beta,Z,P) is the Mittag-Leffler function E_{alpha,beta}(Z) evaluated with accuracy 10^(-P) for each element of Z.

Update 2009-03-25:
(1) Now Z can also be a two-dimensional array.
(2) Addressed the issue reported by Li Jackie. It was caused by rounding errors during computations.

Update 2012-09-07:
(1) Addressed the potential issue reported by Brian Bigler.
(2) Code clean up.

MATLAB release MATLAB 6.0 (R12)
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Comments and Ratings (30)
17 Jun 2014 Pritesh Shah

It is very useful function for implementation of fractional calculus in relativity.
Need small help, E^(3)(0.3,1.9)(-t^0.3).

How do I calculate this function?

17 Feb 2014 qi xiang  
03 Jul 2013 lksdfskl

very good.

02 Jul 2013 Elias Wegert

Positive: The routine computes the function in regions where the Taylor series fails.
Negative: Computations are relatively slow and not completely reliable, for example, alpha=1/2, beta=1 and z near the imaginary axis (probably directly on the imaginary axis) yield completely wrong results (see the phase portrait at http://www.mathe.tu-freiberg.de/~wegert/MittagLefflerBug/Mittag4.png)

23 Jun 2013 Zhuo Li  
28 May 2013 Ondrej

I have one more remark. I found this article on internet:
http://www.bo.infn.it/pinazza/Mainardi/presentazioni/Mainardi/fm60_fcaa_gololu_4.pdf

and it seemed to me that your code is based on it. Probably a reference in your code to this paper (or the theory behind your code), can help other people understand your code.

28 May 2013 Igor Podlubny

To Ondrej: http://mathworld.wolfram.com/Mittag-LefflerFunction.html : we assume alpha > 0, beta > 0. Thank you for the comment on the exception.

22 May 2013 Ondrej

I am not sure, but there might be potential bugs in case a=0.

If a=0, code jumps to the end, at tries to evaluate variable "e"..which does not exist.
Also, as far as I know, case a=0, b=1, corresponds to sum of geometric progression (1/(1-z)), therefore a=0, should be allowed in the code.

13 Sep 2012 wang xiao

Thank you!

12 Sep 2012 Andrey

Thank you Igor!

20 Aug 2012 Brian

This function is excellent and exactly what I needed!

One potential bug:
The compound inequality on line 36 (1<=alf<2) always evaluates to true (1<=alf evaluates to a logical 0 or 1, which is always less than 2). I don't think this was the statement's intent.

20 Aug 2012 Brian  
08 Aug 2012 Jinwen  
12 Jan 2012 Roberto Garrappa

It is a very robust code and is very useful in several applications. An excellent work. It is a reference for practical computation in Fractional Calculus.

31 Mar 2011 udita katugampola

Thanks for the contribution to the field of Fractional Calculus.

20 Nov 2009 Maxwelldemon ??

The code is very very helpful. Thanks a lot. However, there exists a problem:
When 1<alfa<2, beta>1, the returned value has a jump when the variable z is around -170. When alfa=1.5, I tried beta=1.5, 2.5, 3.5, the jump always happens around -170.
Furthermore, in the code, I can't find computations for the case 1<alfa<2, when z>floor(20/(2.1-alfa)^(5.5-2*alfa)).
I will be very thankful for any advice.

12 Apr 2009 Haitao Qi

Thanks. It's the best.

25 Mar 2009 Ivo Petras

Thanks. It works excellent.

10 Oct 2008 tamer nabil  
14 Mar 2008 Li Jackie

Why there exists complex value in the numerical result of the Mittag-Leffler function when the variable (named z) is real?
Recently I used the programe MLF.m to compute the Mittag-Leffler function in Matlab. But I had a problem that there existed complex value in the numerical result of the Mittag-Leffler function. Since the interval of z (variable of function) I setted is a subset of real space, I can't explain why appearence of complex value of Mittag-Leffler function. Could someone give me some advice? Hope it will not delay you too much time. Thanks.

25 Apr 2007 Assabaa Mohamed

can you send me a complete code for simulating the fractional oscillator,I need it urgently to confirm an idea. thanks a lot

23 Mar 2007 Eduardo Cuesta

This function has been useful to me and taking an overview to the code, this seems to be vere, very fine.
Congratulations

09 Oct 2006 Vamsi Krishna

That was very very helpful sir.Please let me know the algorithm you have used or any reference paper which contains the properties you have used.Please let me know if you have written the code for MLF with three parameters.

29 Sep 2006 mervyn yian

may i know do u have a complete code for simulating the fractional oscillator........i need in urgently. thanks a lot

16 Feb 2006 Enrico Scalas

It is a nice and useful algorithm.

15 Feb 2006 saeed alavi

thank a lot .

21 Dec 2005 Alvaro Cartea

It is great to have someone like Podlubny put the code out. This saves people in the field a great deal of time!!

21 Nov 2005 Wang Shaowei

Thank Prof.Igor Podlubny!

12 Nov 2005 Lingzao zeng

Excellent.Thanks a lot

17 Oct 2005 YangQuan Chen

Thanks for the contribution.

Updates
18 Oct 2005

Keywords were truncated by the system during file submission. One can enter many characters in the 'Keywords' field of the file submission form, but it appears that the FileExchange system accepts only first 70 characters in the "Keywords" field.

25 Mar 2009

(1) Now Z can also be a two-dimensional array.
(2) Addressed the issue reported by Li Jackie. It was caused by rounding errors during computations.

07 Sep 2012

Addressed the potential issue reported by Brian Bigler. Code clean up.

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