Two Efficient Generalized Laguerre Spectral Algorithms for Fractional Initial Value Problems

By Dumitru Băleanu, Ali H. Bhrawy, Taha M. Taha. 2013

68 citations recorded by the source corpus.

We present a direct solution technique for approximating linear multiterm fractional differential equations (FDEs) on semi-infinite interval, using generalized Laguerre polynomials. We derive the operational matrix of Caputo fractional derivative of the generalized Laguerre polynomials which is applied together with generalized Laguerre tau approximation for implementing a spectral solution of linear multiterm FDEs on semi-infinite interval subject to initial conditions. The generalized Laguerre pseudo-spectral approximation based on the generalized Laguerre operational matrix is investigated to reduce the nonlinear multiterm FDEs and its initial conditions to nonlinear algebraic system, thus greatly simplifying the problem. Through several numerical examples, we confirm the accuracy and performance of the proposed spectral algorithms. Indeed, the methods yield accurate results, and the exact solutions are achieved for some tested problems.

Subfields: Applied Mathematics, Modeling and Simulation, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/13e2bda0e3b853e781d2646e6bf6212326d5e580

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In Applied Mathematics

In Modeling and Simulation

In Numerical Analysis

Citations in the corpus

A numerical study of fractional relaxation–oscillation equations involving $$\psi $$ψ-Caputo fractional derivative (2018, background): “Next, it was extended to the fractional case by many authors, see for example [3,4,9,10,13,15,24], and the references therein.”
A review of operational matrices and spectral techniques for fractional calculus (2015, methodology): “The generalized Laguerre spectral tau and collocation techniques were studied in [66] for solving linear and nonlinear FDEs on the half line.”
An iterative approach for solving fractional optimal control problems (2018, unknown): “Spectral approximations have been used for solving FDEs by other authors too (see, for instance, Saadatmandi and Dehghan, 2010; Doha et al., 2012; Baleanu et al., 2013; Bhrawy and Zaky, 2015; Bhrawy et al., 2015c; Bhrawy et al. 2015e).”
New spectral techniques for systems of fractional differential equations using fractional-order generalized Laguerre orthogonal functions (2014, methodology): “The operation matrices of fractional derivatives and fractional integrals of generalized Laguerre polynomials were investigated for solving multi-term FDEs on a semi-infinite interval, see [5, 2].”
A unified Petrov-Galerkin spectral method for fractional PDEs (2015, background): “In these applications, fractional partial differential equations (FPDEs) appear of different type such as time- and/or space-fractional diffusion equation of parabolic nature [12], time- and/or space fractional advection, advection–diffusion and Burger’s equations of hyperbolic character [13,14],…”
Polynomial differential quadrature method for numerical solutions of the generalized Fitzhugh–Nagumo equation with time-dependent coefficients (2014, unknown): “According to Shu the stable solution can be obtained when Chebyshev–Gauss–Lobatto grid points [37–43] are chosen.”