New Operational Matrices for Solving Fractional Differential Equations on the Half-Line

Abstract

In this paper, the fractional-order generalized Laguerre operational matrices (FGLOM) of fractional derivatives and fractional integration are derived. These operational matrices are used together with spectral tau method for solving linear fractional differential equations (FDEs) of order ν (0 < ν < 1) on the half line. An upper bound of the absolute errors is obtained for the approximate and exact solutions. Fractional-order generalized Laguerre pseudo-spectral approximation is investigated for solving nonlinear initial value problem of fractional order ν. The extension of the fractional-order generalized Laguerre pseudo-spectral method is given to solve systems of FDEs. We present the advantages of using the spectral schemes based on fractional-order generalized Laguerre functions and compare them with other methods. Several numerical examples are implemented for FDEs and systems of FDEs including linear and nonlinear terms. We demonstrate the high accuracy and the efficiency of the proposed techniques.

A fast second-order parareal solver for fractional optimal control problems

“…The point of departure of the spectral technique is to approximate the solutions yfalse[lfalse]false(x,tfalse) and pfalse[lfalse]false(x,tfalse) of the two PDEs in equations (3) via orthogonal functions false{φjfalse(xfalse)false}j=1m~-1 where m~≥2 is an integer. Here, we consider two kinds of orthogonal functions: the fractional-order generalized Laguerre functions (Bhrawy et al., 2015c) and the shifted Jacobi polynomials (Bhrawy and Zaky, 2015). More choices of the orthogonal functions false{φjfalse(xfalse)false}j=1m~-1 can be found in the work by Bhrawy et al.…”

“…(2015d: Sections 4 and 5), we have which implies that where boldDm~∈ ℝfalse(m~-1false)×false(m~-1false) and Θm~∈ ℝfalse(m~-1false)×false(m~-1false) are the operational matrices of the left Riemann–Liouville fractional integration and the regular second-order derivative, respectively. For the fractional-order generalized Laguerre functions (the shifted Jacobi polynomials), respectively, the concrete expression of the two operational matrices boldDm~ and Θm~ can be found in the work by Bhrawy et al. (2015c) (Bhrawy and Zaky, 2015), respectively.…”

Numerical approach for solving space fractional orderdiffusion equations using shifted Chebyshev polynomials of the fourth kind

“…It should be mentioned that there are also applications from the viewpoint of many fields such as physics, chemistry, engineering, finance, and other sciences that have been developed in the last few decades [12,24,25,27,28,35,43]. A large number of authors studied fractional differential equations, such as [1,6,7,32,39].…”

Using wavelet multi-resolution nature to accelerate the identification of fractional order system

“…The derivation process is similar to that of the linear system. Assuming f (t) = 𝐹 T 𝐻 N (t) and y(t) = 𝑌 T 𝐻 N (t), equation (32) can be simplified by a Haar wavelet operational matrix of integration as follows:…”

“…This method has been used to identify the FOS by Tang et al [27] Li and Sun used the Block-Pulse operational matrix for analyzing the fractional order systems. [28] Other operational matrices include the Legendre operational matrix, [29][30][31][32] Chebyshev operational matrix, [33,34] Jacobi operational matrix, [35] operational matrices of triangular functions, [36] and Haar wavelet operational matrix. [37][38][39] However, for an actual system, the quantity of measurement data is large, which will lead to a high-dimensional operational matrix.…”