**Authors: **Abdulnasir Isah^{1 } & Salisu Ibrahim^{2 }

^{1}Department of Mathematics Education, Faculty of Education, Tishk International University, Erbil, Iraq

^{2}Department of Mathematics Education, Faculty of Education, Tishk International University, Erbil, Iraq

**Abstract:** Genocchi polynomials are known to be defined on the interval [0, 1], but to benefit from the advantages of this polynomials in the field of fractional differential equations (FDEs), it was realized that fractional derivatives of many functions with arbitrary order cannot always be defined at x = 0. To avoid this difficulty, the idea of shifting from the interval [0, 1] to the interval [1, 2] makes it simple and applicable. Interestingly, almost all the properties of the Genocchi polynomials are inherited by the shifted polynomial. Therefore, in this research we construct shifted Genocchi polynomial’s operational matrix of arbitrary order derivative and used it with the collocation method to solve some systems of FDEs.

**Keywords:** Shifted Genocchi Polynomials, Fractional Differential Equations, Operational Matrix, Collocation Method

**References**

Araci, S. (2012). Novel identities for 𝑞-Genocchi numbers and polynomials. Journal of Function Spaces and Applications, 2012.

Araci, S. (2014). Novel identities involving Genocchi numbers and polynomials arising from applications of umbral calculus. Applied Mathematics and Computation, 233, 599-607.

Bayad, A., & Kim, T. (2010). Identities for the Bernoulli, the Euler and the Genocchi numbers and polynomials. Adv. Stud. Contemp. Math, 20(2), 247-253.

Diethelm, K., Ford, N. J., & Freed, A. D. (2002). A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dynamics, 29(1), 3-22.

Isah, A., & Phang, C. (2019). New operational matrix of derivative for solving non-linear fractional differential equations via Genocchi polynomials. Journal of King Saud University-Science, 31(1), 1-7.

Isah, A., & Phang, C. (2017, January). On Genocchi operational matrix of fractional integration for solving fractional differential equations. In AIP Conference Proceedings (Vol. 1795, No. 1, p. 020015). AIP Publishing LLC.

Kilbas, A. A., Srivastava, H. M., & Trujillo, J. J. (2006). Theory and applications of fractional differential equations (Vol. 204). elsevier.

Odibat, Z. (2011). On Legendre polynomial approximation with the VIM or HAM for numerical treatment of nonlinear fractional differential equations. Journal of Computational and Applied Mathematics, 235(9), 2956-2968.

Podlubny, I. (1998). Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Elsevier.

Ray, S. S., Chaudhuri, K. S., & Bera, R. K. (2006). Analytical approximate solution of nonlinear dynamic system containing fractional derivative by modified decomposition method. Applied mathematics and computation, 182(1), 544-552.

Yang, S., Xiao, A., & Su, H. (2010). Convergence of the variational iteration method for solving multi-order fractional differential equations. Computers & Mathematics with Applications, 60(10), 2871-2879.