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.
Highlights
fractional differential equations (FDEs) describe accurately many models in science and engineering such as bioengineering applications, porous or fractured media, electrochemical processes, viscoelastic materials [1,2,3,4,5,6,7]
We extend the application of fractional-order generalized Laguerre collocation (FGLC) method based on fractional-order generalized Laguerre functions (FGLFs) to solve a system of linear FDEs with fractional orders less than 1
The fractional operational matrices of Caputo fractional derivatives and Riemann-Liouville fractional integration were established for these functions
Summary
FDEs describe accurately many models in science and engineering such as bioengineering applications, porous or fractured media, electrochemical processes, viscoelastic materials [1,2,3,4,5,6,7]. Some authors have presented the generalized and modified generalized Laguerre spectral tau and collocation techniques for solving several types of linear and nonlinear FDEs on the half-line, (see [37, 38] and the references therein). We aim to construct the fractional-order generalized Laguerre operational matrices, of fractional derivative and integration, which are used to produce two efficient fractional-order generalized Laguerre tau schemes for solving numerically linear FDEs with initial conditions. We aim to propose a new fractional-order generalized Laguerre collocation (FGLC) scheme for approximating the solution FDE of order ν (0 < ν < 1) with nonlinear terms. The Riemann-Liouville integral Jν f(x) and the Riemann-Liouville fractional derivative Dν f(x) of order ν > 0 are defined by
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