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 f(x). This is achieved by approximating the FIDE using Genocchi polynomials in matrix representation and using the collocation method at equally spaced points within interval [0,1]. This reduces the FIDE into a system of algebraic equations to be solved for the Genocchi coefficients of the solution f(x). 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.

New collocation scheme for solving fractional partial differential equations

“…Other semi-orthogonal polynomials such as Appell polynomials may also had great potential to solve fractional calculus problems effectively. On top of that, we introduced operational matrix of derivative via Genocchi polynomials by Loh et al [10] for the solution of fractional integro-differential equations, (FIDEs), Genocchi operational matrix of integration for solving fractional optimal control problems, (FOCPs) by Phang et al [17] and pantograph equation [7]. Different than these recently developed schemes, here we solve the fractional partial differential equations (FPDEs) by using this Genocchi operational matrix of derivative in conjunction of shifted CGL.…”

Pell-Lucas collocation method to solve high-order linear Fredholm-Volterraintegro-differential equations and residual correction

“…Maleknejad, Mahmoudi, Yalçınbaş and Sezer have studied on the Taylor polynomial approach for linear and nonlinear FVIDEs [21,33]. Also, various methods [7,8,20,29,37,38,40] such as compact the finite difference method [43], the rationalized Haar functions method [22,27], the CAS wavelet method [5], the differential transform method [6], the improved homotopy perturbation method [35], the sine-cosine wavelet method [18,32], the homotopy perturbation method [11,18], the hybrid function method [17], the sinc method [42], the Legendre method [23,25,26,28], the Bernstein method [9,10,36] and the combined Laplace transform-Adomian decomposition method [41] have been studied for solving linear and nonlinear FVIDEs.…”

Approximate solution of linear Volterra integro-differential equation by using cubic B-spline finite element method in the complex plane

“…We can solve integro-differential equations with some basis functions by the Haar and rationalized function methods [4][5][6][7][8], Adomian decomposition method [9], Legendre wavelets method [10], RBFs collocation method [11], and Genocchi polynomials and collocation method based on the Bernoulli operational matrix [12,13]. Also, in [14], Volterra-Fredholm integro-differential equations of fractional order are solved by the sinc-collocation method.…”