A New Operational Matrix of Fractional Derivatives to Solve Systems of Fractional Differential Equations via Legendre Wavelets

Abstract

This paper introduces a new numerical approach to solving a system of fractional differential equations (FDEs) using the Legendre wavelet operational matrix method (LWOMM). We first formulated the operational matrix of fractional derivatives in some special conditions using some notable characteristics of Legendre wavelets and shifted Legendre polynomials. Then, the system of fractional differential equations was transformed into a system of algebraic equations by using these operational matrices. At the end of this paper, several examples are presented to illustrate the effectivity and correctness of the proposed approach. Comparing the methodology with several recognized methods demonstrates that the advantages of the Legendre wavelet operational matrix method are its accuracy and the understandability of the calculations.

The Gegenbauer Wavelets-Based Computational Methods for the Coupled System of Burgers’ Equations with Time-Fractional Derivative

“…where F 2 is the Gegenbauer wavelet coefficients vector. Now by substituting Equations (26), (32)-(34), (41) and (42) into Equations (24) and (32), (33), (35), and (41)-(43) into Equation (25), respectively, then using operational matrices of integration, we get the residuals functions R 1 (x, t) and R 2 (x, t) for this system as follows:…”

Caputo fractional derivative operational matrices of Legendre and Chebyshev wavelets in fractional delay optimal control

“…in which the required initial conditionẋ(0) =ẋ 0 is replaced with the final condition (68). Since in the method of using derivative operation, we see in (32) that unlike the integral operation there is no arising term about the initial condition, so we must add the given constraint to the QP model of the new system (39), (40).…”

“…By selecting this k, the dilation property of wavelets has been removed [11]. There is a similar statement for derivative operational matrices of Legendre wavelets; in [5,40], the operational matrices are constructed for shifted Legendre polynomials by choosing k = 0. An obvious indication is that the solutions were presented in one interval while the wavelets solutions are piecewise-defined functions that are presented in two (at least) or more subintervals and because of this, we consider the compatibility constraint in our QP formulations.…”

Two Reliable Methods for The Solution of Fractional Coupled Burgers’ Equation Arising as a Model of Polydispersive Sedimentation

“…Up to now, various powerful numerical techniques have been proposed for solutions of the (FPDEs). Some of them are, Adomian decomposition method (ADM) [21,23], homotopy perturbation method (HPM) [2,14], variational iteration method (VIM) [22], Legendre wavelet operational matrix method (LWOMM) [26], homotopy analysis method (HAM) [18,19] and residual power series method [6,7].…”