The main objective of this paper is to present weighted $k$ $k$ -fractional integral and derivative operators of a function with respect to another function and to uncover their properties. In addition to this, we find the weighted Laplace transform of the newly defined operators. As applications of the weighted $k$ $k$ -fractional operators in mathematical physics, we study the fractional forms of kinetic differintegral equation and the time-fractional heat equation involving the novel operators and find their solutions using weighted Laplace transform.

Keywords:

References

1. H. Laurent , Sur le calcul des derivees a indicies quelconques, Nouv. Ann. Math. 3 (1884) 240–252. Google Scholar
2. N. Y. Sonin , On differentiation with arbitrary index, Moscow Mat. Sb. 6 (1869) 1–38. Google Scholar
3. A. V. Letnikov , Theory of differentiation with an arbitrary index, Moscow Mat. Sb. 3 (1868) 1–66 (in Russian). Google Scholar
4. D. Zhao and M. Luo , Representations of acting processes and memory effects: General fractional derivative and its application to theory of heat conduction with finite wave speeds, Appl. Math. Comput. 346 (2019) 31–544. Crossref, Web of Science, Google Scholar
5. G. Dattoli, L. Gianessi, L. Mezi, D. Tocci and R. Colai , FEL time-evolution operator, Nucl. Instrum. Methods Phys. Res. A 304 (1991) 541–544. Crossref, Web of Science, Google Scholar
6. N. Abel , Solution de quelques problémes á l’aide dintégrales définies, Mag. Naturv. 2 (1823) 1–27. Google Scholar
7. R. Herrmann , Fractional Calculus: An Introduction for Physicists, 2nd edn. (World Scientific, New Jersey, 2014). Link, Google Scholar
8. R. Hilfer , Applications of Fractional Calculus In Physics (World Scientific, New Jersey, 2000). Link, Google Scholar
9. R. Hilfer , Threefold Introduction to Fractional Derivatives (Anomalous Transport: Foundations and Applications, 2008). Crossref, Google Scholar
10. T. Du, H. Wang, M. A. Khan and Y. Zhang , Certain integral inequalities considering generalized $m$ $m$ -convexity on fractal sets and their applications, Fractals 27(7) (2019) 1950117. Link, Web of Science, Google Scholar
11. J. A. Machado , And i say to myself: “What a fractional world!”, Fract. Calc. Appl. Anal. 14(4) (2011) 635–654. Crossref, Web of Science, Google Scholar
12. M. A. Khan and T. U. Khan , Parameterized Hermite–Hadamard type inequalities for fractional integrals, Turkish J. Ineq. 1(1) (2017) 26–37. Google Scholar
13. H. Yaldz and A. O. Akdemir , Katugampola fractional integrals within the class of convex functions, Turkish J. Sci. 3(1) (2018) 40–50. Google Scholar
14. A. Ekinci and M. E. Özdemir , Some new integral inequalities via Riemann–Liouville integral operators, Appl. Comput. Math. 18(3) (2019) 288–295. Web of Science, Google Scholar
15. E. Set, A. O. Akdemir and F. Ozata , Grüss type inequalities for fractional integral operator involving the extended generalized Mittag-Leffler function, Appl. Comput. Math. 19(3) (2020) 402–414. Web of Science, Google Scholar
16. E. Set, A. O. Akdemir and E. Ozdemir , Simpson type integral inequalities for convex functions via Riemann–Liouville integrals, Filomat 31(14) (2017) 4415–4420. Crossref, Web of Science, Google Scholar
17. E. Set, J. Choi and B. Elik , Certain Hermite–Hadamard type inequalities involving generalized fractional integral operators, Rev. R Acad. Cienc. Exactas Fs. Nat., Ser. A Mat. 112(4) (2018) 1539–1547. Crossref, Web of Science, Google Scholar
18. S. G. Samko, A. A. Kilbas and O. I. Marichev , Fractional Integrals and Derivatives: Theory and Applications (Gordon and Breach Science Publishers, 1993). Google Scholar
19. M. Dalir and M. Bashour , Applications of fractional calculus, Appl. Math. Sci. 4 (2010) 1021–1032. Google Scholar
20. K. Miller and B. Ross , An Introduction to the Fractional Calculus and fractional Differential Equations (John Wiley and Sons, 1993). Google Scholar
21. T. O. Salim and A. W. Faraj , A Generalization of Mittag-Leffler function and integral operator associated with integral calculus, J. Frac. Calc. Appl. 3 (2012) 1–13. Google Scholar
22. H. M. Srivastava and Z. Tomovski , Fractional calculus with an integral operator containing generalized Mittag-Leffler function in the kernal, Appl. Math. Comput. 211 (2009) 198–210. Crossref, Web of Science, Google Scholar
23. Z. Tomovski, R. Hiller and H. M. Srivastava , Fractional and operational calculus with generalized fractional derivative operators and Mittag-Leffler function, Integral Transform. Spec. Funct. 21 (2011) 797–814. Crossref, Web of Science, Google Scholar
24. M. Samraiz, Z. Perveen, T. Abdeljawad, S. Iqbal and S. Naheed , On certain Fractional Calculus operators and their applications, Phys. Scr. 95(11) (2020) 115210. Crossref, Web of Science, Google Scholar
25. M. Samraiz, Z. Perveen, T. Abdeljawad, G. Rahman, K. S. Nisa and D. Kumar , On $(k, s)$ $(k, s)$ -Hilfer-Prabhakar fractional derivative with applications in mathematical physics, Front. Phys. 8 (2020) 309. Crossref, Web of Science, Google Scholar
26. H. M. Srivastava and R. K. Saxena , Operators of fractional integration and their applications, App. Math. Comp. 118(1) (2001) 1–52. Crossref, Web of Science, Google Scholar
27. A. A. Kilbas, H. M. Srivastava and J. J. Trujillo , Theory and Applications of Fractional Differential Equations (Elsevier, Amsterdam, 2006). Google Scholar
28. O. Agarwal , Some generalized fractional calculus operators and their applications in integral equations, Fract. Calc. Appl. Anal. 15(4) (2012) 700–711. Crossref, Web of Science, Google Scholar
29. R. Almeida , A Caputo fractional derivative of a function with respect to another function, Commun. Nonlinear Sci. Numer. Simul. 44 (2017) 460–481. Crossref, Web of Science, Google Scholar
30. F. Jarad and T. Abdeljawad , Generalized fractional derivatives and Laplace transform, Discrete Contin. Dyn. Syst., Ser. S. 13(3) (2020) 709–722. Web of Science, Google Scholar
31. V. Kiryakova , A brief story about the operators of generalized fractional calculus, Fract. Calc. Appl. Anal. 11(2) (2008) 203–220. Google Scholar
32. T. J. Osler , Fractional derivatives of a composite function, SIAM J. Math. Anal. 1(2) (1970) 288–293. Crossref, Google Scholar
33. M. Al-Refai and A. M. Jarrah , Fundamental results on weighted Caputo–Fabrizio fractional derivative, Chaos Solitons Fractals 126 (2019) 7–11. Crossref, Web of Science, Google Scholar
34. M. Al-Refai , On weighted Atangana–Baleanu fractional operators, Adv. Differ. Equ. 2020 (2020) 3. Crossref, Web of Science, Google Scholar
35. G. A. Dorrego and R. A. Cerutti , The $k$ $k$ -Mittag-Leffler Function, Int. J. Contemp. Math. Sci. 7(15) (2012) 705–716. Google Scholar
36. R. Diaz and E. Pariguan , On hypergeometric functions and Pochhammer $k$ $k$ -symbol. Divulg. Math. 15(2) (2007) 179–192. Google Scholar
37. F. Jarad, T. Abdeljawad and K. Shah , On the weighted fractional of a function with respect to another function, Fractals 28(8) (2020) 2040011. Link, Web of Science, Google Scholar
38. G. A. Dorrego , Generalized Riemann–Liouville fractional operators associated with a generalization of the Prabhakar integral operator Progr. Fract. Differ. Appl. 2(2) (2016) 131–140. Crossref, Google Scholar
39. S. K. Panchal, P. V. Dole and A. D. Khandagale , $k$ $k$ -Hilfer-Prabhakar fractional derivatives and its applications, Indian J. Math. 59(3) (2017) 367–383. Google Scholar
40. R. Hilfer and L. Anton , Fractional master equations and fractal time random walks, Phys. Rev. E. 51(2) (1995) 848–851. Crossref, Web of Science, Google Scholar
41. R. Hilfer , Applications of Fractional Calculus in Physics (World Scientific, London, 2000). Link, Google Scholar
42. Y. Luchko , Some uniqueness and existence results for the initial-boundary-value problems for the generalized time-fractional diffusion equation, Comput. Math. Appl. 59(5) (2010) 1766–1772. Crossref, Web of Science, Google Scholar
43. Y. Luchko , Initial-boundary-value problems for the generalized multi-term time-fractional diffusion equation, J. Math. Anal. Appl. 374(2) (2011) 538–548. Crossref, Web of Science, Google Scholar