An Intriguing Approach to The Fractional Mellin Transform Method

Keywords: Mellin transform, fractional differential equation, the CTIT transformation, fractional derivative


We propose an adapted Mellin transform method that gives the solution of a fractional di¤erential equation with variable coefficients in ordinary domain. After we mention a transformation
of cosmic time to individual time (CTIT), we explain how it can reduce the problem from fractional form to ordinary form when it is used with Mellin transformation, via an example for 0 < alpha < 1; where alpha is the order of fractional derivative. Then, we give an application of the results.


1. Kılıçman, A, A note on Mellin transform and distributions, Mathematical & Computational Applications, ,
No. 1, pp. 65‐72, 2004.
2. Y. Zhang, H. M. Srivastava and M. C. Baleanu, Local fractional variational iteration algorithm II for nonhomogeneous
model associated with the non-differentiable heat flow, Advances in Mechanical Engineering,
2015, Vol. 7(10) 1‐5.
3. A. Alsaedi, J. J. Nieto and V. Venktesh, Fractional electrical circuits, Advances in Mechanical Engineering, 2015,
Vol. 7(12) 1‐7.
4. A. Atangana, J. J. Nieto, Numerical solution for the model of RLC circuit via the fractional derivative without
singular kernel, Advances in Mechanical Engineering, 2015, Vol. 7(10) 1‐7.
5. A. Atangana, B. S. Alkahtani, Extension of the resistance, inductance, capacitance electrical circuit to fractional
derivative without singular kernel, Advances in Mechanical Engineering, 2015, Vol. 7(6) 1‐6.
6. Q. Gao, Y. Hou, T. Deng, C. Wang and R. Hou, Extended state observer based fractional order
proportional−integral−derivative controller for a novel electro-hydraulic servo system with -actuation
balancing and positioning, Advances in Mechanical Engineering, 2015, Vol. 7(12) 1‐11.
7. Q. Gao, K. Li, Y. Hou, R. Hou and C. Wang, Balancing and positioning for a gun control system based on fuzzy
fractional order proportional−integral−derivative strategy, Advances in Mechanical Engineering, 2016, Vol. 8(3)
8. A. M. Lopes, J. T. Machado, Visualizing control systems performance: A fractional perspective, Advances in
Mechanical Engineering, 2015, Vol. 7(12) 1‐8.
9. Y. Zhang, S. Wang, G. Liu and J.Yang, Computer-aided diagnosis of abnormal breasts in mammogram images
by weighted‐type fractional Fourier transform, Advances in Mechanical Engineering, 2016, Vol. 8(2) 1‐11.
10. F. AlAzemi, A. AlAzemi, L. Boyadjiev, Mellin transform method for solving the Black‐ Scholes Equation,
International Journal of Pure and Applied Mathematics, Volume 97 No. 32014, 287‐301.
11. Magorzata Klimek, Daniel Dziembowski, On Mellin transform application to solution of fractional dierential
equations, Scientic Research of the Institute of Mathematics and Computer Science, 2008, Volume 7, Issue 2,
pages 31‐42.
12. O. Gonz’alez‐Gaxiola, Jos’e A. Santiago, An ‐Mellin Transform and Some of its Applica‐ tions, Int. J. Contemp.
Math. Sciences, Vol. 7, 2012, no. 48, 2353‐2361.
13. Yanka Nikolova, ‐Mellin Transform and One of Its Applications, Mathematica Balkanica New Series Vol. 26,
2012, Fasc. 1‐2.
14. I. Podlubny, Geometric and physical interpretation of fractional integration and fractional differentiation, Fractional
Calculus and Applied Analysis 5 (4) (2002) 367‐386.
15. E. Demirci, N. Özalp, A method for solving differential equations of fractional order, J. Comput. Appl. Math. 236,
2754‐2762 (2012).
16. I. Koca, A method for solving dierential equations of ‐fractional order, Applied Mathematics and Computation
266 (2015) 1‐5.
17. Katugampola, U. N., Mellin Transforms of Generalized Fractional Integrals and Derivatives, Applied Mathematics
and Computation 00 (2014) 1‐17.
18. Adem Kılıçman, Maryam Omran, Note on fractional Mellin transform and applications, SpringerPlus (2016) 5:100
DOI 10.1186/s40064-016-1711-x.
19. I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999.
20. Nuri Ozalp, Ozlem Ozturk Mizrak, Fractional Laplace transform method in the framework of the CTIT
transformation, Journal of Computational and Applied Mathematics (2016), DOI: 10.