Some properties of Sadik transform and its applications of fractional-order dynamical systems in control theory

Year 2020, Volume: 4 Issue: 1, 51 - 66, 31.03.2020

Saleh S. Redhwan Sadikali L. Shaikh Mohammed S. Abdo

https://doi.org/10.31197/atnaa.647503

Cited By: 7

Abstract

In this paper, we study some new properties of Sadik transform such as integration, time delay, initial value theorem, and final value theorem. Moreover, we prove the theorem of Sadik transform for Caputo fractional derivative and we also establish sufficient conditions for the existence of the Sadik transform of Caputo fractional derivatives. At the end, the fractional-order dynamical systems in control theory as an application of this transform is discussed, in addition, some numerical examples to justify our results.

Keywords

A fractional differential equation, , Caputo fractional derivative, Integral transforms, Sadik transform

References

• 1- G.E. Andrews, R. Askey, R. Roy, Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 71 (1999).
• 2- P. L. Butzer, U. Westphal, An Introduction to Fractional Calculus, World Scientific. Singapore. View at Zentralblatt MATH. (2000).
• 3- H. Eltayeb, A K\i l\i \c{c}man, On some applications of a new integral transform, Int. J. Math. Anal. Analysis. 4 (2010) 123-132.
• 4- G. Jumarie, Laplace's transform of fractional order via the Mittag-Leffler function and modified Riemann-Liouville derivative, Appl. Math. Lett. 22 (2009) 1659-1664.
• 5- P. Jigen, Laplace transform and fractional differential equations, Appl. Math. Lett. 24 (2011) 2019-2023.
• 6- L. Kexue, P. Jigen, Laplace transform and fractional differential equations, Appl. Math. Lett. 24 (2011) 2019-2023.
• 7- A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Math. Stud Elsevier Amsterdam, 204 (2006).
• 8- A. Kili\c{c}man, H. Eltayeb, On the applications of Laplace and Sumudu transforms, J. Franklin Inst. 347 (2010) 848--862.
• 9- A. Kili\c{c}man, H. Eltayeb, A note on integral transforms and partial differential equations, Appl. Math. Sci. 4 (2010) 109-118.
• 10- K. S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley \& Sons, New York, NY, USA. (1993).
• 11- N. S Abhale, S. S Pawar, Fundamental properties of Sadik transform and it's applications, J. Appl. Sci. Comput. 6 (2019) 995-999.
• 12- K. B. Oldham, J. Spanier, The Fractional Calculus, Academic Press, New York, NY, USA. (1974).
• 13- I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Elsevier, 198 (1999).
• 14- G. D. Medina, N. R. Ojeda, J. H Pereira, Fractional Laplace transform and fractional calculus, int. math. forum, 12 (2017) 991-1000.
• 15- S. L. Shaikh, M. S. Chaudhary, On a new integral transform and solution of some integral equations IJPAM, 73 (2011) 299- 308.
• 16- S. L. Shaikh, Introducing a new integral transform Sadik transform, Amer. Int. J. Res. Sci. Tech. Eng. Math. (2018) 100-102.
• 17- S. L. Shaikh, Sadik transform in control theory, Int. J. Innov. Sci. Res. Tech. 3 (2018) 1-3.
• 18- S. L. Shaikh, Some applications of the new integral transform for Partial differential Equations, math.jour.interdisciplinary sci. 7 (2018) 45-49.
• 19- M. E. Tarig, S. M. Elzaki, E. A. Elnour, on the new integral transform Elzaki transform fundamental properties investigations and applications, Global J. Math. Sci. Theory Pract. 4 (2012) 1-13.