The Ramanujan Journal

, Volume 48, Issue 1, pp 21–32

# Generalization of Taylor’s formula and differential transform method for composite fractional q-derivative

• Lata Chanchlani
• Subhash Alha
• Jaya Gupta
Article

## Abstract

In the present paper, we first establish a generalized q-Taylor’s formula involving composite fractional q-derivative. Next, we define the generalized q-differential transform and its inverse for composite fractional q-derivative and establish some basic properties for this transform. We also illustrate the effectiveness of these results by solving two fractional q-difference equations.

## Keywords

q-Taylor’s formula Composite fractional q-Derivative q-Differential transform method q-Difference equation

## Mathematics Subject Classification

26A33 33E12 34A12 35A22 33D90

## Notes

### Acknowledgements

The support provided through UGC-Minor Research Project under XII plan grant of Maulana Azad National Urdu University, Hyderabad is gratefully acknowledged. The authors are grateful to Prof. Mridula Garg for inspiring discussions and helpful comments during the preparation of the paper. The authors are also thankful to the anonymous referee for the fruitful suggestions which led to the present form of paper.

## References

1. 1.
Agarwal, R.P.: Certain fractional $$q$$-integrals and $$q$$-derivatives. Proc. Camb. Philos. Soc. 66, 365–370 (1969)
2. 2.
Al-Salam, W.A.: Some fractional $$q$$-integrals and $$q$$-derivatives. Proc. Edinb. Math. Soc. 15, 135–140 (1996)
3. 3.
Annaby, M.H., Mansour, Z.S.: $$q$$-Fractional Calculus and Equations. Springer, Heidelberg (2012)
4. 4.
Garg, M., Chanchlani, L., Alha, S.: On generalized $$q$$-differential transform. Aryabhatt J. Math. Inform. 5(2), 265–274 (2013)
5. 5.
Hassan, H.A.: Generalized q-Taylor formula. Adv. Differ. Equ. 1, 1–2 (2016)
6. 6.
Hilfer, R.: Application of Fractional Calculus in Physics. World Scientific Publishing, Singapore (2000)
7. 7.
Jing, S.C., Fan, H.Y.: $$q$$-Taylor’s formula with its $$q$$-remainder. Commun. Theor. Phys. 23(1), 117–120 (1995)
8. 8.
Odibat, Z., Momani, S., Erturk, V.S.: Generalized differential transform method: application to differential equations of fractional order. Appl. Math. Comput. 197, 467–477 (2008)
9. 9.
Rajković, P.M., Marinković, S.D., Stanković, M.S.: On $$q$$-analogues of Caputo derivatives and Mittag-Leffler function. Fract. Calc. Appl. Anal. 10(4), 359–374 (2007)
10. 10.
Rajković, P.M., Marinković, S.D., Stanković, M.S.: Fractional integrals and derivatives in $$q$$-calculus. Appl. Anal. Discret. Math. 1, 311–323 (2007)
11. 11.
Zhou, J.K.: Differential Transformation and Its Applications for Electrical Circuits. Huazhong University Press, Wuhan (1986)Google Scholar

## Authors and Affiliations

• Lata Chanchlani
• 1
• Subhash Alha
• 2
• Jaya Gupta
• 3
1. 1.Department of Science and HumanitiesGovt. R. C. Khaitan Polytechnic CollegeJaipurIndia