March  2020, 13(3): 797-804. doi: 10.3934/dcdss.2020045

Note on a $ k $-generalised fractional derivative

1. 

The IIS University, Jaipur, Rajasthan-302020, India

2. 

Manipal University Jaipur, Rajasthan-303007, India

Corresponding author: ekta.jaipur@gmail.com; sunil.joshi@jaipur.manipal.edu

Received  May 2018 Revised  August 2018 Published  March 2019

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.

Citation: Ekta Mittal, Sunil Joshi. Note on a $ k $-generalised fractional derivative. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 797-804. doi: 10.3934/dcdss.2020045
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. GarraR. GorenfloF. 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. MridulaP. ManoharL. 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. MubeenA. Rehman and F. Shaheen, Properties of k-gamma, k-beta and k-psi functions, Bothalia Journal, 5 (2014), 371-379. 

[12]

L. G. RomeroL. L. LuqueG. 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. GarraR. GorenfloF. 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. MridulaP. ManoharL. 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. MubeenA. Rehman and F. Shaheen, Properties of k-gamma, k-beta and k-psi functions, Bothalia Journal, 5 (2014), 371-379. 

[12]

L. G. RomeroL. L. LuqueG. 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.

[1]

Haisheng Tan, Liuyan Liu, Hongyu Liang. Total $\{k\}$-domination in special graphs. Mathematical Foundations of Computing, 2018, 1 (3) : 255-263. doi: 10.3934/mfc.2018011

[2]

Yu-Ming Chu, Saima Rashid, Fahd Jarad, Muhammad Aslam Noor, Humaira Kalsoom. More new results on integral inequalities for generalized $ \mathcal{K} $-fractional conformable Integral operators. Discrete and Continuous Dynamical Systems - S, 2021, 14 (7) : 2119-2135. doi: 10.3934/dcdss.2021063

[3]

Silvia Frassu. Nonlinear Dirichlet problem for the nonlocal anisotropic operator $ L_K $. Communications on Pure and Applied Analysis, 2019, 18 (4) : 1847-1867. doi: 10.3934/cpaa.2019086

[4]

Jianqin Zhou, Wanquan Liu, Xifeng Wang, Guanglu Zhou. On the $ k $-error linear complexity for $ p^n $-periodic binary sequences via hypercube theory. Mathematical Foundations of Computing, 2019, 2 (4) : 279-297. doi: 10.3934/mfc.2019018

[5]

Yasemin Cengellenmis, Abdullah Dertli, Steven T. Dougherty, Adrian Korban, Serap Şahinkaya, Deniz Ustun. Reversible $ G $-codes over the ring $ {\mathcal{F}}_{j,k} $ with applications to DNA codes. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2021056

[6]

Yishui Wang, Dongmei Zhang, Peng Zhang, Yong Zhang. Local search algorithm for the squared metric $ k $-facility location problem with linear penalties. Journal of Industrial and Management Optimization, 2021, 17 (4) : 2013-2030. doi: 10.3934/jimo.2020056

[7]

Adam Kanigowski, Federico Rodriguez Hertz, Kurt Vinhage. On the non-equivalence of the Bernoulli and $ K$ properties in dimension four. Journal of Modern Dynamics, 2018, 13: 221-250. doi: 10.3934/jmd.2018019

[8]

Chenchen Wu, Wei Lv, Yujie Wang, Dachuan Xu. Approximation algorithm for spherical $ k $-means problem with penalty. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021067

[9]

Xavier Gràcia, Xavier Rivas, Narciso Román-Roy. Erratum: Constraint algorithm for singular field theories in the $ k $-cosymplectic framework. Journal of Geometric Mechanics, 2021, 13 (2) : 273-275. doi: 10.3934/jgm.2021007

[10]

Huaning Liu, Yixin Ren. On the pseudorandom properties of $ k $-ary Sidel'nikov sequences. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2021038

[11]

Min Li, Yishui Wang, Dachuan Xu, Dongmei Zhang. The approximation algorithm based on seeding method for functional $ k $-means problem. Journal of Industrial and Management Optimization, 2022, 18 (1) : 411-426. doi: 10.3934/jimo.2020160

[12]

Habibul Islam, Om Prakash, Ram Krishna Verma. New quantum codes from constacyclic codes over the ring $ R_{k,m} $. Advances in Mathematics of Communications, 2022, 16 (1) : 17-35. doi: 10.3934/amc.2020097

[13]

David Hoff. Pointwise bounds for the Green's function for the Neumann-Laplace operator in $ \text{R}^3 $. Kinetic and Related Models, 2022, 15 (4) : 535-550. doi: 10.3934/krm.2021037

[14]

Huimin Zheng, Xuejun Guo, Hourong Qin. The Mahler measure of $ (x+1/x)(y+1/y)(z+1/z)+\sqrt{k} $. Electronic Research Archive, 2020, 28 (1) : 103-125. doi: 10.3934/era.2020007

[15]

Fan Yuan, Dachuan Xu, Donglei Du, Min Li. An exact algorithm for stable instances of the $ k $-means problem with penalties in fixed-dimensional Euclidean space. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021122

[16]

Augusto Visintin. $ \Gamma $-compactness and $ \Gamma $-stability of the flow of heat-conducting fluids. Discrete and Continuous Dynamical Systems - S, 2022, 15 (8) : 2331-2343. doi: 10.3934/dcdss.2022066

[17]

Pak Tung Ho. Prescribing $ Q $-curvature on $ S^n $ in the presence of symmetry. Communications on Pure and Applied Analysis, 2020, 19 (2) : 715-722. doi: 10.3934/cpaa.2020033

[18]

Alexander A. Davydov, Stefano Marcugini, Fernanda Pambianco. Upper bounds on the length function for covering codes with covering radius $ R $ and codimension $ tR+1 $. Advances in Mathematics of Communications, 2022  doi: 10.3934/amc.2021074

[19]

Yong Xia, Ruey-Lin Sheu, Shu-Cherng Fang, Wenxun Xing. Double well potential function and its optimization in the $N$ -dimensional real space-part Ⅱ. Journal of Industrial and Management Optimization, 2017, 13 (3) : 1307-1328. doi: 10.3934/jimo.2016074

[20]

Shu-Cherng Fang, David Y. Gao, Gang-Xuan Lin, Ruey-Lin Sheu, Wenxun Xing. Double well potential function and its optimization in the $N$ -dimensional real space-part Ⅰ. Journal of Industrial and Management Optimization, 2017, 13 (3) : 1291-1305. doi: 10.3934/jimo.2016073

2020 Impact Factor: 2.425

Metrics

  • PDF downloads (353)
  • HTML views (598)
  • Cited by (1)

Other articles
by authors

[Back to Top]