Note on the fractional Mittag-Leffler functions by applying the modified Riemann-Liouville derivatives
DOI:
https://doi.org/10.5269/bspm.44103Abstract
In this article, the fractional derivatives in the sense of the modified Riemann-Liouville derivative is employed for constructing some results related to Mittag-Leffler functions and established a number of important relationships between the Mittag-Leffler functions and Wright function.
References
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15. Letnikov, A. V., Theory of differentiation of fractional order, Mat. Sb 3, 1, 1868, (1868).
16. Lu, B., Backlund transformation of fractional Riccati equation and its applications to nonlinear fractional partial differential equations, Physics Letters A, 376, 2045-2048, (2012). https://doi.org/10.1016/j.physleta.2012.05.013
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18. Miller, K. S. and Ross, B., An introduction to the fractional calculus and fractional differential equations, (1993).
19. Jumarie, G., Stochastic differential equations with fractional Brownian motion input, International journal of systems science 24, 6, 1113-1131, (1993). https://doi.org/10.1080/00207729308949547
20. Jumarie, G., Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results, Computers and Mathematics with Applications 51, 9-10, 1367-1376, (2006). https://doi.org/10.1016/j.camwa.2006.02.001
21. Jumarie, G., Table of some basic fractional calculus formulae derived from a modified Riemann-Liouville derivative for non-differentiable functions, Applied Mathematics Letters 22, 3, 378-385, (2009). https://doi.org/10.1016/j.aml.2008.06.003
22. Prabhakar, T. R., A singular integral equation with a generalized Mittag Leffler function in the kernel, (1971).
23. S. Zhang, S. and Zhang, H. Q. Fractional sub-equation method and its applications to nonlinear fractional PDEs, Physics Letters Section A, 375, 7, 1069-1073, (2011). https://doi.org/10.1016/j.physleta.2011.01.029
2. Al-Akaidi, Marwan. Fractal speech processing, Cambridge university press, (2004). https://doi.org/10.1017/CBO9780511754548
3. Bayram, M., Adiguzel, H. and Secer, A., Oscillation criteria for nonlinear fractional differential equation with damping term, Open Physics, 14, 1, 119-128, (2016). https://doi.org/10.1515/phys-2016-0012
4. Bayram, M., Secer, A., and Adiguzel, H., On the oscillation of fractional order nonlinear differential equations, Sakarya University Journal of Science, 21, 6, 1512-1523, (2017).
5. Campos, L. M. B. C., On a concept of derivative of complex order with applications to special functions, IMA Journal of Applied Mathematics 33, 2, 109-133, (1984). https://doi.org/10.1093/imamat/33.2.109
6. Caputo, M., Linear models of dissipation whose Q is almost frequency independent-II, Geophysical Journal International 13, 5, 529-539, (1967). https://doi.org/10.1111/j.1365-246X.1967.tb02303.x
7. Erdelyi, A., Asymptotic Expansions, Dover Publications, New York, (1954). https://doi.org/10.21236/AD0055660
8. Erdelyi, A. (Ed). Tables of Integral Transforms. vol. 1, McGraw-Hill, (1954).
9. Faraz, N., Khan, Y., Jafari, H., Yildirim, A. and Madani, M., Fractional variational iteration method via modified Riemann-Liouville derivative, Journal of King Saud University, 23 , 4), 413-417, (2011). https://doi.org/10.1016/j.jksus.2010.07.025
10. Kiryakova, V., Unified approach to univalency of the Dziok-Srivastava and the fractional calculus operators, Advances in Mathematics 1, 33-43, (2012).
11. Kiryakova, V., Criteria for univalence of the Dziok-Srivastava and the Srivastava-Wright operators in the class A, Applied Mathematics and Computation 218.3, 883-892, (2011). https://doi.org/10.1016/j.amc.2011.01.076
12. Kiryakova, V., The multi-index Mittag-Leffler functions as an important class of special functions of fractional calculus, Computers & Mathematics with Applications 59, 5, 1885-1895, (2010). https://doi.org/10.1016/j.camwa.2009.08.025
13. Kiryakova, V., The operators of generalized fractional calculus and their action in classes of univalent functions, Geometric Function Theory and Applications 29, 40, (2010).
14. Kiryakova, V. and Saigo M., TO PRESERVE UNIVALENCY OF ANALYTIC FUNCTIONS, Comptes rendus de l'Acad'emie bulgare des Sciences 58, 10, 1127-1134, (2005).
15. Letnikov, A. V., Theory of differentiation of fractional order, Mat. Sb 3, 1, 1868, (1868).
16. Lu, B., Backlund transformation of fractional Riccati equation and its applications to nonlinear fractional partial differential equations, Physics Letters A, 376, 2045-2048, (2012). https://doi.org/10.1016/j.physleta.2012.05.013
17. Luchko, Yuri. Operational method in fractional calculus, Fract. Calc. Appl. Anal 2, 4, 463-488, (1999).
18. Miller, K. S. and Ross, B., An introduction to the fractional calculus and fractional differential equations, (1993).
19. Jumarie, G., Stochastic differential equations with fractional Brownian motion input, International journal of systems science 24, 6, 1113-1131, (1993). https://doi.org/10.1080/00207729308949547
20. Jumarie, G., Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results, Computers and Mathematics with Applications 51, 9-10, 1367-1376, (2006). https://doi.org/10.1016/j.camwa.2006.02.001
21. Jumarie, G., Table of some basic fractional calculus formulae derived from a modified Riemann-Liouville derivative for non-differentiable functions, Applied Mathematics Letters 22, 3, 378-385, (2009). https://doi.org/10.1016/j.aml.2008.06.003
22. Prabhakar, T. R., A singular integral equation with a generalized Mittag Leffler function in the kernel, (1971).
23. S. Zhang, S. and Zhang, H. Q. Fractional sub-equation method and its applications to nonlinear fractional PDEs, Physics Letters Section A, 375, 7, 1069-1073, (2011). https://doi.org/10.1016/j.physleta.2011.01.029
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2021-12-18
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