New Operational Matrix via Genocchi Polynomials for Solving Fredholm-Volterra Fractional Integro-Differential Equations

Abstract

It is known that Genocchi polynomials have some advantages over classical orthogonal polynomials in approximating function, such as lesser terms and smaller coefficients of individual terms. In this paper, we apply a new operational matrix via Genocchi polynomials to solve fractional integro-differential equations (FIDEs). We also derive the expressions for computing Genocchi coefficients of the integral kernel and for the integral of product of two Genocchi polynomials. Using the matrix approach, we further derive the operational matrix of fractional differentiation for Genocchi polynomial as well as the kernel matrix. We are able to solve the aforementioned class of FIDE for the unknown function <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>. This is achieved by approximating the FIDE using Genocchi polynomials in matrix representation and using the collocation method at equally spaced points within interval <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mo stretchy="false">[</mml:mo><mml:mn fontstyle="italic">0,1</mml:mn><mml:mo stretchy="false">]</mml:mo></mml:math>. This reduces the FIDE into a system of algebraic equations to be solved for the Genocchi coefficients of the solution <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M3"><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>. A few numerical examples of FIDE are solved using those expressions derived for Genocchi polynomial approximation. Numerical results show that the Genocchi polynomial approximation adopting the operational matrix of fractional derivative achieves good accuracy comparable to some existing methods. In certain cases, Genocchi polynomial provides better accuracy than the aforementioned methods.