Sadik transform and some result in fractional calculus

S.S. Redhwan; S.L. Shaikh; M.S. Abdo; S.Y. Al-Mayyahi

doi:10.26637/MJM0802/0037

Abstract

In this research paper, we present some new properties of Sadik transform which are related to the fractional calculus including Reimann-Liouville fractional operator, then we prove new results of Sadik transform like the infinite series, the convolution theorem and the Mittag-Leffler function. Moreover, it is shown that the Sadik transform method is an efficient technique for obtaining an exact analytic solution of some linear fractional differential equations. Some numerical examples to justify our results are illustrated.

Keywords:

Integral transforms, fractional derivative and fractional integral, Sadik transform.

Mathematics Subject Classification:

Mathematics

Authors Imprint References Funding Related Articles Downloads

S.S. Redhwan Research Scholar, Department of Mathematics, Dr. Babasaheb Ambedkar Marathwada University , Aurangabad-431004, India. https://orcid.org/0000-0002-4779-7657
S.L. Shaikh Department of Mathematics, Maulana Azad College of arts, Science and Commerce, RozaBagh, Aurangabad-431001, Maharashtra, India. https://orcid.org/0000-0003-3360-9069
M.S. Abdo Department of Mathematics, Hodeidah University, Al-Hodeidah, Yemen.
S.Y. Al-Mayyahi Department of Mathematics, College of Education of Pure Science, University of Wasit, Iraq.

Pages: 536-543
Date Published: 01-04-2020
Vol. 8 No. 02 (2020): Malaya Journal of Matematik (MJM)

S. Abbas, M. Benchohra and G. M. N'Guérékata,Topics in Fractional Differential Equations, Springer Science, Business Media, 27(2012).

M. S. Abdo and S. K. Panchal, Existence and continuous dependence for fractional neutral functional differential equations, Journal of Mathematical Modeling, 5(2)(2017), 153-170.

M. S. Abdo and S. K. Panchal, Weighted fractional neutral functional differential equations. Journal of Siberian Federal University Mathematics & Physics, 11 (2018), 535-549.

B. Ahmad and J. Nieto, Existence results for nonlinear boundary value problems of fractional integrodifferential equations with integral boundary conditions, Boundary Value Problems. 2009(1)(2009). 708576.

R. Agarwal, M. Benchohra and S. Hamani, boundary value problems for fractional differential equations, Georgian Math. J., 16(3)(2009), 401-411.

P. L. Butzer and U. Westphal, An Introduction to Fractional Calculus, World Scientific. Singapore. View at Zentralblatt MATH. (2000).

G. Jumarie, Laplace's transform of fractional order via the Mittag-Leffler function and modified Riemann-Liouville derivative, Applied Mathematics Letters , 22(2009), 16591664.

${ }^{[8]}$ L. Kexue and P. Jigen, Laplace transform and fractional differential equations.Appl. Math. Lett., 24 (2011), 20192023.

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Math. Stud Elsevier Amsterdam, $204(2006)$.

A. Kiliçman and H. Eltayeb, On the applications of Laplace and Sumudu transforms, J. Franklin Inst., 347 (2010), 848-862.

A. Kiliçman and H. Eltayeb, A note On Integral Transforms and Partial Differential Equations, App. Math. Sci., $4(2010), 109-118$.

K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, NY, USA. (1993).

K. B. Oldham and J. Spanier,The Fractional Calculus, Academic Press, New York, NY, USA. (1974).

I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Elsevier, 198 (1998).

G. D. Medina, N. R. Ojeda, J. H. Pereira and L. G Romero, Fractional Laplace Transform and Fractional Calculus, Int. Math. Forum , 12(2017), 991-1000.

S. L. Shaikh, Introducing a new integral transform Sadik transform, Amer. Int. J. Res. Sci. Tech. Eng. Math, (2018), 100-102.

S. L. Shaikh and M. S. Chaudhary, On a new integral transform and solution of some integral equations ,IJPAM , 73(2011), 299-308.

S. L. Shaikh, Sadik transform in control theory, Int. J. Innov. Sci. Res. Tech, 3(2018), 1-3.

S. L. Shaikh, Some applications of the new integral transform for Partial differential Equations, Math. Jour. Interdisciplinary Sci. , 7 (2018), 45-49.

S. Saleh Redhwan, L. Sadikali Shaikh, and S. Mohammed Abdo, Some properties of Sadik transform and its applications of fractional-order dynamical systems in control theory, Advances in the Theory of Nonlinear Analysis and its Application , 4(1) (2019), 51-66.