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Fractional Solutions of a k-hypergeometric Differential Equation

Year 2019, Volume: 2 Issue: 3, 212 - 214, 30.12.2019

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.

References

  • [1] K. S. Miller, and B. Ross, An introduction to the fractional calculus and fractional differential equations, Wiley, 1993.
  • [2] K. Oldham, and J. Spanier, The fractional calculus theory and applications of differentiation and integration to arbitrary order, Elsevier, 1974.
  • [3] I. Podlubny, Matrix approach to discrete fractional calculus. Fractional calculus and applied analysis, 3(4) (2000), 359-386.
  • [4] H. T. Michael, The Laplace transform in discrete fractional calculus, Computers and Mathematics with Applications 62(3) (2011) 1591-1601.
  • [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] P. T. Kuchment, Floquet theory for partial differential equations, Birkhäuser, 2012.
  • [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] P. Verdonck, The role of computational fluid dynamics for artificial organ design, Artificial organs 26(7) (2002), 569-570.
  • [9] A. Mandelis, Diffusion-wave fields: mathematical methods and Green functions, Springer Science and Business Media, 2013.
  • [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] 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] P. Flajolet, Combinatorial aspects of continued fractions, Discrete mathematics 306(10-11) (2006), 992-1021.
  • [13] G. Plonka, D. Potts, G. Steidi, M. Tasche, Fourier series, Numerical Fourier Analysis, Birkhäuser, Cham, 1-59, 2018.
  • [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] 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] R. Yilmazer, and O. Ozturk, On Nabla discrete fractional calculus operator for a modified Bessel equation, Therm. Sci. 22 (2018) 203-209.
  • [17] R. Yilmazer, N-fractional calculus operator N-method to a modified hydrogen atom equation, Mathematical Communications 15(2) (2010), 489-501.
  • [18] R. Yilmazer, Discrete fractional solutions of a Hermite equation, Journal of Inequalities and Special Functions, 10(1) (2019), 53-59.
  • [19] R. Yilmazer, Discrete fractional solution of a non-Homogeneous non-Fuchsian differential equations, Thermal Science, 23(1) (2019), 121-127.
  • [20] R. Yilmazer, M. Inc, and M. Bayram, On discrete fractional solutions of Non-Fuchsian differential equations, Mathematics 6(12) (2018), 308.
  • [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] L. Shengfeng, and Y. Dong, k-Hypergeometric series solutions to one type of non-homogeneous k-Hypergeometric Equations, Symmetry 11(2) (2019), 262.
  • [23] E. E. Kummer, De integralibus quibusdam definitis et seriebus infinitis, Journal für die reine und angewandte Mathematik 17 (1837), 228-242.
  • [24] L. Campos, On some solutions of the extended confluent hypergeometric differential equation, Journal of computational and applied mathematics 137(1) (2001), 177-200.
Year 2019, Volume: 2 Issue: 3, 212 - 214, 30.12.2019

Abstract

References

  • [1] K. S. Miller, and B. Ross, An introduction to the fractional calculus and fractional differential equations, Wiley, 1993.
  • [2] K. Oldham, and J. Spanier, The fractional calculus theory and applications of differentiation and integration to arbitrary order, Elsevier, 1974.
  • [3] I. Podlubny, Matrix approach to discrete fractional calculus. Fractional calculus and applied analysis, 3(4) (2000), 359-386.
  • [4] H. T. Michael, The Laplace transform in discrete fractional calculus, Computers and Mathematics with Applications 62(3) (2011) 1591-1601.
  • [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] P. T. Kuchment, Floquet theory for partial differential equations, Birkhäuser, 2012.
  • [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] P. Verdonck, The role of computational fluid dynamics for artificial organ design, Artificial organs 26(7) (2002), 569-570.
  • [9] A. Mandelis, Diffusion-wave fields: mathematical methods and Green functions, Springer Science and Business Media, 2013.
  • [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] 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] P. Flajolet, Combinatorial aspects of continued fractions, Discrete mathematics 306(10-11) (2006), 992-1021.
  • [13] G. Plonka, D. Potts, G. Steidi, M. Tasche, Fourier series, Numerical Fourier Analysis, Birkhäuser, Cham, 1-59, 2018.
  • [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] 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] R. Yilmazer, and O. Ozturk, On Nabla discrete fractional calculus operator for a modified Bessel equation, Therm. Sci. 22 (2018) 203-209.
  • [17] R. Yilmazer, N-fractional calculus operator N-method to a modified hydrogen atom equation, Mathematical Communications 15(2) (2010), 489-501.
  • [18] R. Yilmazer, Discrete fractional solutions of a Hermite equation, Journal of Inequalities and Special Functions, 10(1) (2019), 53-59.
  • [19] R. Yilmazer, Discrete fractional solution of a non-Homogeneous non-Fuchsian differential equations, Thermal Science, 23(1) (2019), 121-127.
  • [20] R. Yilmazer, M. Inc, and M. Bayram, On discrete fractional solutions of Non-Fuchsian differential equations, Mathematics 6(12) (2018), 308.
  • [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] L. Shengfeng, and Y. Dong, k-Hypergeometric series solutions to one type of non-homogeneous k-Hypergeometric Equations, Symmetry 11(2) (2019), 262.
  • [23] E. E. Kummer, De integralibus quibusdam definitis et seriebus infinitis, Journal für die reine und angewandte Mathematik 17 (1837), 228-242.
  • [24] L. Campos, On some solutions of the extended confluent hypergeometric differential equation, Journal of computational and applied mathematics 137(1) (2001), 177-200.
There are 24 citations in total.

Details

Primary Language English
Subjects Engineering
Journal Section Articles
Authors

Resat Yilmazer 0000-0002-5059-3882

Karmina K. Ali This is me 0000-0002-3815-4457

Publication Date December 30, 2019
Acceptance Date December 27, 2019
Published in Issue Year 2019 Volume: 2 Issue: 3

Cite

APA Yilmazer, R., & Ali, K. K. (2019). Fractional Solutions of a k-hypergeometric Differential Equation. Conference Proceedings of Science and Technology, 2(3), 212-214.
AMA Yilmazer R, Ali KK. Fractional Solutions of a k-hypergeometric Differential Equation. Conference Proceedings of Science and Technology. December 2019;2(3):212-214.
Chicago Yilmazer, Resat, and Karmina K. Ali. “Fractional Solutions of a K-Hypergeometric Differential Equation”. Conference Proceedings of Science and Technology 2, no. 3 (December 2019): 212-14.
EndNote Yilmazer R, Ali KK (December 1, 2019) Fractional Solutions of a k-hypergeometric Differential Equation. Conference Proceedings of Science and Technology 2 3 212–214.
IEEE R. Yilmazer and K. K. Ali, “Fractional Solutions of a k-hypergeometric Differential Equation”, Conference Proceedings of Science and Technology, vol. 2, no. 3, pp. 212–214, 2019.
ISNAD Yilmazer, Resat - Ali, Karmina K. “Fractional Solutions of a K-Hypergeometric Differential Equation”. Conference Proceedings of Science and Technology 2/3 (December 2019), 212-214.
JAMA Yilmazer R, Ali KK. Fractional Solutions of a k-hypergeometric Differential Equation. Conference Proceedings of Science and Technology. 2019;2:212–214.
MLA Yilmazer, Resat and Karmina K. Ali. “Fractional Solutions of a K-Hypergeometric Differential Equation”. Conference Proceedings of Science and Technology, vol. 2, no. 3, 2019, pp. 212-4.
Vancouver Yilmazer R, Ali KK. Fractional Solutions of a k-hypergeometric Differential Equation. Conference Proceedings of Science and Technology. 2019;2(3):212-4.