| Title:
|
Note on a discretization of a linear fractional differential equation (English) |
| Author:
|
Čermák, Jan |
| Author:
|
Kisela, Tomáš |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
135 |
| Issue:
|
2 |
| Year:
|
2010 |
| Pages:
|
179-188 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
The paper discusses basics of calculus of backward fractional differences and sums. We state their definitions, basic properties and consider a special two-term linear fractional difference equation. We construct a family of functions to obtain its solution. (English) |
| Keyword:
|
fractional difference |
| Keyword:
|
fractional sum |
| Keyword:
|
discrete Mittag-Leffler function |
| MSC:
|
26A33 |
| MSC:
|
39A12 |
| idZBL:
|
Zbl 1224.39003 |
| idMR:
|
MR2723085 |
| DOI:
|
10.21136/MB.2010.140695 |
| . |
| Date available:
|
2010-07-20T18:35:37Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/140695 |
| . |
| Reference:
|
[1] Atici, F. M., Eloe, P. W.: A transform method in discrete fractional calculus.Int. J. Difference Equ. 2 (2007), 165-176. MR 2493595 |
| Reference:
|
[2] Atici, F. M., Eloe, P. W.: Initial value problems in discrete fractional calculus.Proc. Amer. Math. Soc. 137 (2009), 981-989. Zbl 1166.39005, MR 2457438, 10.1090/S0002-9939-08-09626-3 |
| Reference:
|
[3] Bohner, M., Peterson, A.: Dynamic Equations on Time Scales.An Introduction with Applications, Birkhäuser, Boston, MA (2001). Zbl 0993.39010, MR 1843232 |
| Reference:
|
[4] Čermák, J., Nechvátal, L.: On $(q,h)$-analogue of fractional calculus.J. Nonlinear Math. Phys. 17 (2010), 1-18. Zbl 1189.26006, MR 2647460, 10.1142/S1402925110000593 |
| Reference:
|
[5] Gray, H. L., Zhang, N. F.: On a new definition of the fractional difference.Math. Comp. 50 (1988), 513-529. Zbl 0648.39002, MR 0929549, 10.1090/S0025-5718-1988-0929549-2 |
| Reference:
|
[6] Miller, K. S., Ross, B.: Fractional Difference Calculus.Proc. Int. Symp. Unival. Funct., Frac. Calc. Appl., Koriyama, Japan, May 1988, 139-152; Ellis Horwood Ser. Math. Appl., Horwood, Chichester, 1989. Zbl 0693.39002, MR 1199147 |
| Reference:
|
[7] Miller, K. S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations.John Wiley & Sons, New York (1993). Zbl 0789.26002, MR 1219954 |
| Reference:
|
[8] Díaz, R., Teruel, C.: $q,k$-Generalized Gamma and Beta Functions.J. Nonlin. Math. Phys. 12 (2005), 118-134. Zbl 1075.33010, MR 2122869, 10.2991/jnmp.2005.12.1.10 |
| Reference:
|
[9] Díaz, J. B., Osler, T. J.: Differences of fractional order.Math. Comp. 28 (1974), 185-202. MR 0346352, 10.2307/2005825 |
| Reference:
|
[10] Podlubný, I.: Fractional Differential Equations.Academic Press, San Diego (1999). MR 1658022 |
| . |
