Fractional Solutions of a k-hypergeometric Differential Equation

Abstract

In the present work, we study the second order homogeneous $k$-hypergeometric differential equation by utilizing the discrete fractional Nabla calculus operator. As a result, we obtained a novel exact fractional solution to the given equation.

Keywords

• Discrete fractional,the k-hypergeometric differential equation,Nabla operator.

Kaynakça

1. [1] K. S. Miller, and B. Ross, An introduction to the fractional calculus and fractional differential equations, Wiley, 1993.
2. [2] K. Oldham, and J. Spanier, The fractional calculus theory and applications of differentiation and integration to arbitrary order, Elsevier, 1974.
3. [3] I. Podlubny, Matrix approach to discrete fractional calculus. Fractional calculus and applied analysis, 3(4) (2000), 359-386.
4. [4] H. T. Michael, The Laplace transform in discrete fractional calculus, Computers and Mathematics with Applications 62(3) (2011) 1591-1601.
5. [5] M. N. Özi¸sik, H. R. B. Orlande, M. J. Colac, R. M. Cotta, Finite difference methods in heat transfer, CRC press, 2017.
6. [6] P. T. Kuchment, Floquet theory for partial differential equations, Birkhäuser, 2012.
7. [7] A. H. Khater, M. H. M. Moussa, and S. F. Abdul-Aziz, Invariant variational principles and conservation laws for some nonlinear partial differential equations with variable coefficients part II, Chaos, Solitons and Fractals 15(1) (2013), 1-13.
8. [8] P. Verdonck, The role of computational fluid dynamics for artificial organ design, Artificial organs 26(7) (2002), 569-570.

9. [9] A. Mandelis, Diffusion-wave fields: mathematical methods and Green functions, Springer Science and Business Media, 2013.
10. [10] G. M. Viswanathan, The hypergeometric series for the partition function of the 2D Ising model, Journal of Statistical Mechanics: Theory and Experiment 2015(7) (2015), 07004.
11. [11] C. M. Bender, C. B. Dorje, and P.M. Markus, Hamiltonian for the zeros of the Riemann zeta function, Physical Review Letters 118(13) (2017), 130201.
12. [12] P. Flajolet, Combinatorial aspects of continued fractions, Discrete mathematics 306(10-11) (2006), 992-1021.
13. [13] G. Plonka, D. Potts, G. Steidi, M. Tasche, Fourier series, Numerical Fourier Analysis, Birkhäuser, Cham, 1-59, 2018.
14. [14] J. W. Cooley, J. W. Tukey, An algorithm for the machine calculation of complex fourier series, Mathematics of computation, 19(90), 297-301, 1965.
15. [15] R. Yilmazer, M. Inc, F. Tchier, D. Baleanu, Particular solutions of the confluent hypergeometric differential equation by using the nabla fractional calculus operator, Entropy 18(2) (2016), 49.
16. [16] R. Yilmazer, and O. Ozturk, On Nabla discrete fractional calculus operator for a modified Bessel equation, Therm. Sci. 22 (2018) 203-209.
17. [17] R. Yilmazer, N-fractional calculus operator N-method to a modified hydrogen atom equation, Mathematical Communications 15(2) (2010), 489-501.
18. [18] R. Yilmazer, Discrete fractional solutions of a Hermite equation, Journal of Inequalities and Special Functions, 10(1) (2019), 53-59.
19. [19] R. Yilmazer, Discrete fractional solution of a non-Homogeneous non-Fuchsian differential equations, Thermal Science, 23(1) (2019), 121-127.
20. [20] R. Yilmazer, M. Inc, and M. Bayram, On discrete fractional solutions of Non-Fuchsian differential equations, Mathematics 6(12) (2018), 308.
21. [21] M. Inc and R. Yilmazer, On some particular solutions of the Chebyshev’s equation by means of Ña discrete fractional calculus operator, Prog. Fract. Differ. Appl. 2(2) (2016), 123-129.
22. [22] L. Shengfeng, and Y. Dong, k-Hypergeometric series solutions to one type of non-homogeneous k-Hypergeometric Equations, Symmetry 11(2) (2019), 262.
23. [23] E. E. Kummer, De integralibus quibusdam definitis et seriebus infinitis, Journal für die reine und angewandte Mathematik 17 (1837), 228-242.
24. [24] L. Campos, On some solutions of the extended confluent hypergeometric differential equation, Journal of computational and applied mathematics 137(1) (2001), 177-200.