The novel operational matrices based on 2D-Genocchi polynomials: solving a general class of variable-order fractional partial integro-differential equations

Abstract

The main purpose of this study was to introduce an efficient approximate approach for solving a general class of variable-order fractional partial integro-differential equations (VO-FPIDEs). First, the Genocchi polynomials properties and the pseudo-operational matrix of the VO-fractional derivative and fractional integration are presented. Then, to approximate an integral part of the problems, we obtain the dual pseudo-operational matrix of fractional order with a new technique. The pseudo-operational matrices of fractional order and Genocchi collocation method are applied to reduce the VO-FPIDEs to a system of algebraic equations. The error estimation indicates that the approximate solution converges to the exact solution. Also, we discuss in detail on the upper bound of error for the pseudo-operational matrix of fractional integration. In addition, to illustrate the efficiency and applicability of the approach, we present several numerical examples. Keywords Genocchi polynomials • Fractional pseudo-operational matrix • Variable-order fractional partial integro-differential equations • Mixed Riemann-Liouville integral • Variable-order fractional derivative Mathematics Subject Classification 26A33 • 33F05 • 35R09 Communicated by Agnieszka Malinowska.

A Spectral Approach for Time-Fractional Diffusion and Subdiffusion Equations in a Large Interval

“…where g k denotes the Genocchi numbers. Now, we mention some significant properties of Genocchi polynomials [12]:…”

“…Any function g defined over the interval [0, 1] can be expanded in terms of GPs which is described in detail in [12].…”

“…This section investigates the error bound of the integral POM and the error of partial derivative of the approximate solution. To compute the upper bound of error for POM of integration with respect to variable x, we use Equation (3.1), Lemmas 1 and 2 in [12], and the following relation:…”

An Accurate Numerical Scheme for Three-Dimensional Variable-Order Time-Fractional Partial Differential Equations in Two Types of Space Domains

“…In the meantime, numerical methods for solving VO-fractional differential equations worked very powerfully. Therefore, several numerical methods have been introduced, such as fractionalorder Taylor wavelets [24], spline finite difference scheme [13], modified wavelet method [6], Genocchi collocation method [5], etc. This paper presents the numerical framework based on discrete shifted Hahn polynomials for solving 3D-VO time-fractional partial differential equations.…”

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

“…The Genocchi polynomials, G n ðxÞ, is one of the members of the Appell polynomials, A n ðxÞ, satisfying the differential relation ðdA n ðxÞÞ/dx = nA n−1 ðxÞ, n = 1, 2, 3, ⋯. Besides, many new results are obtained in the field of number theory and combinatory [1][2][3][4], the Genocchi polynomials are also applied successfully to solve some kind of fractional calculus problems, and its advantages were described in [5][6][7][8][9] mostly via its operational matrix. However, most of the results are applied over the interval ½0, 1.…”