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Note on a $ k $-generalised fractional derivative
| 1. | The IIS University, Jaipur, Rajasthan-302020, India |
| 2. | Manipal University Jaipur, Rajasthan-303007, India |
In this paper, we introduce the $ k $-generalised fractional derivatives with three parameters which reduced to $ k $-fractional Hilfer derivatives and $ k $-Riemann-Liouville fractional derivative as an interesting special cases. Further, we have also introduced some presumably new fascinating results which include the image power function, Laplace transform and composition of $ k $-Riemann-Liouville fractional integral with generalized composite fractional derivative. The technique developed in this paper can be used in other situation as well.
References:
| [1] |
R. Dıaz and E. Pariguan,
On hypergeometric functions and pochhammer k-symbol, Divulg. Mat, 15 (2007), 179-192.
|
| [2] |
G. A. Dorrego and R. A. Cerutti,
The k-fractional hilfer derivative, International Journal of Mathematical Analysis, 7 (2013), 543-550.
doi: 10.12988/ijma.2013.13051. |
| [3] |
R. Garra, R. Gorenflo, F. Polito and Ž Tomovski,
Hilfer–prabhakar derivatives and some applications, Applied Mathematics and Computation, 242 (2014), 576-589.
doi: 10.1016/j.amc.2014.05.129. |
| [4] |
R. Hilfer et al., Applications of Fractional Calculus in Physics, vol. 35, World Scientific, 2000.
doi: 10.1142/9789812817747. |
| [5] |
O. S. Iyiola,
Solving k-fractional hilfer differential equations via combined fractional integral transform methods, British Journal of Mathematics & Computer Science, 4 (2014), 1427-1436.
doi: 10.9734/BJMCS/2014/9444. |
| [6] |
C. G. Kokologiannaki,
Properties and inequalities of generalized k-gamma, beta and zeta functions, Int. J. Contemp. Math. Sciences, 5 (2010), 653-660.
|
| [7] |
M. Mansour,
Determining the k-generalized gamma function $\gamma$k (x) by functional equations, Int. J. Contemp. Math. Sciences, 4 (2009), 1037-1042.
|
| [8] |
K. S. Miller and B. Ross, An introduction to the fractional calculus and fractional differential equations, A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1993. |
| [9] |
G. Mridula, P. Manohar, L. Chanchalani and A. Subhash,
On generalized composite fractional derivative, Walailak Journal of Science and Technology (WJST), 11 (2014), 1069-1076.
|
| [10] |
S. Mubeen and G. Habibullah,
k-fractional integrals and application, Int. J. Contemp. Math. Sci, 7 (2012), 89-94.
|
| [11] |
S. Mubeen, A. Rehman and F. Shaheen,
Properties of k-gamma, k-beta and k-psi functions, Bothalia Journal, 5 (2014), 371-379.
|
| [12] |
L. G. Romero, L. L. Luque, G. A. Dorrego and R. A. Cerutti,
On the k-riemann-liouville fractional derivative, Int. J. Contemp. Math. Sci, 8 (2013), 41-51.
doi: 10.12988/ijcms.2013.13004. |
show all references
References:
| [1] |
R. Dıaz and E. Pariguan,
On hypergeometric functions and pochhammer k-symbol, Divulg. Mat, 15 (2007), 179-192.
|
| [2] |
G. A. Dorrego and R. A. Cerutti,
The k-fractional hilfer derivative, International Journal of Mathematical Analysis, 7 (2013), 543-550.
doi: 10.12988/ijma.2013.13051. |
| [3] |
R. Garra, R. Gorenflo, F. Polito and Ž Tomovski,
Hilfer–prabhakar derivatives and some applications, Applied Mathematics and Computation, 242 (2014), 576-589.
doi: 10.1016/j.amc.2014.05.129. |
| [4] |
R. Hilfer et al., Applications of Fractional Calculus in Physics, vol. 35, World Scientific, 2000.
doi: 10.1142/9789812817747. |
| [5] |
O. S. Iyiola,
Solving k-fractional hilfer differential equations via combined fractional integral transform methods, British Journal of Mathematics & Computer Science, 4 (2014), 1427-1436.
doi: 10.9734/BJMCS/2014/9444. |
| [6] |
C. G. Kokologiannaki,
Properties and inequalities of generalized k-gamma, beta and zeta functions, Int. J. Contemp. Math. Sciences, 5 (2010), 653-660.
|
| [7] |
M. Mansour,
Determining the k-generalized gamma function $\gamma$k (x) by functional equations, Int. J. Contemp. Math. Sciences, 4 (2009), 1037-1042.
|
| [8] |
K. S. Miller and B. Ross, An introduction to the fractional calculus and fractional differential equations, A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1993. |
| [9] |
G. Mridula, P. Manohar, L. Chanchalani and A. Subhash,
On generalized composite fractional derivative, Walailak Journal of Science and Technology (WJST), 11 (2014), 1069-1076.
|
| [10] |
S. Mubeen and G. Habibullah,
k-fractional integrals and application, Int. J. Contemp. Math. Sci, 7 (2012), 89-94.
|
| [11] |
S. Mubeen, A. Rehman and F. Shaheen,
Properties of k-gamma, k-beta and k-psi functions, Bothalia Journal, 5 (2014), 371-379.
|
| [12] |
L. G. Romero, L. L. Luque, G. A. Dorrego and R. A. Cerutti,
On the k-riemann-liouville fractional derivative, Int. J. Contemp. Math. Sci, 8 (2013), 41-51.
doi: 10.12988/ijcms.2013.13004. |
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