Numerical Solution for Arbitrary Domain of Fractional Integro-differential Equation via the General Shifted Genocchi Polynomials

Abstract

The Genocchi polynomial has been increasingly used as a convenient tool to solve some fractional calculus problems, due to their nice properties. However, like some other members in the Appell polynomials, the nice properties are always limited to the interval defined in $\left[0,1\right]$ . In this paper, we extend the Genocchi polynomials to the general shifted Genocchi polynomials, $S_n^{\left(a,b\right)}\left(x\right)$ , which are defined for interval $\left[a,b\right]$ . New properties for this general shifted Genocchi polynomials will be introduced, including the determinant form. This general shifted Genocchi polynomials can overcome the conventional formula of finding the Genocchi coefficients of a function $f\left(x\right)$ that involves $f^{\left(n-1\right)}\left(x\right)$ which may not be defined at $x=0,1$ . Hence, we use the general shifted Genocchi polynomials to derive the operational matrix and hence to solve the Fredholm-type fractional integro-differential equations with arbitrary domain $\left[a,b\right]$ .