Katugampola kinetic fractional equation with its solution

Abstract

The purpose of this research is to investigate the result of Katugampola kinetic fractional equations containing the first kind of generalized Bessel's function. This paper considers the manifold generality of the first kind generalized Bessel's function in form of the solution of Katugampola kinetic fractional equations. The $\tau$ Laplace transform technique is used to obtain the result. In addition, a graphical representation is included for viewing the behavior of the gained solutions.

Keywords

• Laplace transform
• Katugampola fractional operator
• kinetic equation

Kaynakça

1. [1] T. Abdeljawad, On conformable fractional calculus, J. Comput. Appl. Math. 279 (2013) 57-66.
2. [2] T. Abdeljawad, S. Rashid, Z. Hammouch, Y.M. Chu, Some new local fractional inequalities associated with generalized (s,m)-convex functions and applications, Adv. Differ. Equ. 2020(1) (2020) 1-27.
3. [3] P. Agarwal, M. Chand, G. Singh, Kinetic fractional equations involving generalized k-Bessel function via Sumudu transform, Alex. Eng. J. 55(4) (2016) 3053-3059.
4. [4] Á. Baricz, Generalized Bessel Functions of the First Kind, Vol. 1994 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 2010.
5. [5] Á. Baricz, Geometric properties of generalized Bessel func-tions, Publicationes Mathematicae Debrecen. 73(1-2) (2008) 155-178.
6. [6] D. Baleanu, P. Agarwal, S.D. Purohit, Certain fractional integral formulas involving the product of generalized Bessel functions, Sci. World J. 2013 (2013) Article ID 567132 9 pages.
7. [7] S.B. Chen, S. Rashid, M.A. Noor, Z. Hammouch, Y.M. Chu, New fractional approaches for n-polynomial P-convexity with applications in special function theory, Adv. Differ. Equ. 2020(1) (2020) 1-31.
8. [8] A. Chouhan, S. Sarswat, On solution of generalized Kinetic equation of fractional order, Int. j. math. sci. appl. 2(2) (2012) 813-818.

9. [9] V.B.L. Chaurasia, D. Kumar, On the solution of generalized kinetic fractional equation, Adv. Stud. Theor. Phys. 4 (2010) 773-780.
10. [10] V.B.L. Chaurasia, S.C. Pandey, On the new computable solution of the generalized kinetic fractional equations involving the generalized function for the fractional calculus and related functions, Astrophys. Space Sci. 317 (2008) 213-219.
11. [11] J. Choi, D. Kumar, Solutions of generalized kinetic fractional equations involving Aleph functions, Math. Commun. 20 (2015) 113-123.
12. [12] G. Dorrego, D. Kumar, A generalization of the kinetic equation using the Prabhakar-type operators, Honam Math. J. 39(3) (2017) 401- 416.
13. [13] B.K. Dutta, L.K. Arora, J. Borah, On the solution of kinetic fractional equation, Gen. Math. Notes 6 (2011) 40-48 .
14. [14] L. Galué, A generalized Bessel function, Integral Transforms Spec. Funct. 14(5) (2003) 395-401 .
15. [15] V.G. Gupta, B. Sharma, F.B.M. Belgacem, On the solutions of generalized kinetic fractional equations, Appl. Math. Sci. 5(17-20) (2011) 899-910.
16. [16] H.J. Haubold, A.M. Mathai, The kinetic fractional equation and thermonuclear functions, Astrophys, Space Sci. 273 (2000) 53-63.
17. [17] F. Jarad, T. Abdeljawad, A modified Laplace transform for certain generalized fractional operators, Results in Nonlinear Analysis. 1(2) (2018) 88-98.
18. [18] M. Kamarujjama, N.U. Khan, O. Khan, The generalized p-k-Mittag-Leffler function and solution of kinetic fractional equations, J. Anal. In press. https://doi.org/10.1007/s41478-018-0160-z.
19. [19] U.N. Katugampola, New approach to a generalized fractional integral, Appl. Math. Comput. 218(3) (2011) 860-865.
20. [20] D. Kumar, J. Choi, H.M. Srivastava, Solution of a general family of kinetic fractional equations associated with the generalized Mittag- Leffler function, Nonlinear Funct. Anal. Appl. 23(3) (2018) 455-471.
21. [21] D. Kumar, S.D. Purohit, A. Secer, A. Atangana, On generalized kinetic fractional equations involving generalized Bessel function of the first kind, Math. Probl. Engg. 2015 (2015) Article ID 289387, 7 pages.
22. [22] G. Mittag-Leffler, Sur la representation analytique d’une branche uniforme d’une fonction monogene, Acta Math. 29(1) (1905) 101-181.
23. [23] S. Rashid, Z. Hammouch, H. Kalsoom, R. Ashraf, Y.M. Chu, New investigation on the generalized K-fractional integral operators, Front. Phys. 8 (2020) 25.
24. [24] A.I. Saichev, G.M. Zaslavsky, Kinetic fractional equations: solutions and applications, Chaos. 7(4) (1997) 753-764.
25. [25] R.K. Saxena, A.M. Mathai, H.J. Haubold, On kinetic fractional equations, Astrophys Space Sci. 282 (2002) 281-287.
26. [26] R.K. Saxena, A.M. Mathai, H.J. Haubold, On generalized kinetic fractional equations, Physica A. 344 (2004) 657-664.
27. [27] R.K. Saxena, S.L. Kalla, On the solutions of certain kinetic fractional equations, Appl. Math. Comput. 199 (2008) 504-511.
28. [28] G.N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press; The Macmillan, Cambridge, UK, 1944.