Solutions of Singular Differential Equations by means of Discrete Fractional Analysis

Abstract

Recently, many researchers demonstrated the usefulness of fractional calculus in the derivation of particular solutions of linear ordinary and partial differential equation of the second order. In this study, we acquire new discrete fractional solutions of singular differential equations (homogeneous and nonhomogeneous) by using discrete fractional nabla operator ${{\nabla }^{\upsilon }}(0<\upsilon <1).$

Keywords

• Discrete fractional analysis,Nabla operator,Singular differential equations.

References

1. [1] K. S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley and Sons, Inc., New York, 1993.
2. [2] I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999.
3. [3] C. Goodrich, A. C. Peterson, Discrete Fractional Calculus, Berlin: Springer, 2015.
4. [4] R. Hilfer (Ed.), Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.
5. [5] H. L., Gray, N., Zhang, On a New Definition of the Fractional Difference, Mathematics of Computation, 50 (182) (1988), 513-529.
6. [6] F. M. Atici, P.W. Eloe, Discrete fractional calculus with the nabla operator, Electronic Journal of Qualitative Theory of Differential Equations, Spec. Ed I, 3 (2009), 1-12.
7. [7] N. Acar, F. M. Atici, Exponential functions of discrete fractional calculus, Appl. Anal. Discrete Math. 7 (2013), 343-353.
8. [8] G. A. Anastassiou, Right nabla discrete fractional calculus, Int. J. Difference Equations, 6 (2011), 91-104.

9. [9] J. J. Mohan, Analysis of nonlinear fractional nabla difference equations, Int. J. Analysis Applications 7 (2015), 79-95.
10. [10] R. Yilmazer, et al., Particular Solutions of the Confluent Hypergeometric Differential Equation by Using the Nabla Fractional Calculus Operator, Entropy, 18 (49) (2016), 1-6.
11. [11] R. Yilmazer, O. Ozturk, On Nabla Discrete Fractional Calculus Operator for a Modified Bessel Equation, Therm. Sci., 22 (2018), S203-S209.
12. [12] R. Yilmazer, Discrete fractional solution of a Hermite Equation, Journal of Inequalities and Special Functions, 10 (1) (2019), 53-59.
13. [13] R. Yilmazer, Discrete fractional solution of a non-homogeneous non-fuchsian differential equations, Therm. Sci., 23 (2019), 121-127.
14. [14] R. Yilmazer, $N-$fractional calculus operator ${{N}^{\mu }}$ method to a modified hydrogen atom equation, Math. Commun., 15 (2010), 489-501.
15. [15] W. G. Kelley, A. C. Peterson, Difference Equations: An Introduction with Applications, Academic Press, San Diego, 2001.