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New operational matrices for solving fractional differential equations on the half-line.

Bhrawy AH, Taha TM, Alzahrani EO, Baleanu D, Alzahrani AA - PLoS ONE (2015)

Bottom Line: 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.

View Article: PubMed Central - PubMed

Affiliation: Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia; Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef, Egypt.

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.

No MeSH data available.


Graph of the absolute error function for N = 10, α = 0 and ν = λ = 0.5, for Example 6.
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pone.0126620.g003: Graph of the absolute error function for N = 10, α = 0 and ν = λ = 0.5, for Example 6.

Mentions: In Table 7, we list the results obtained by the fractional-order generalized Laguerre generalized collocation (FGLC) method with various choices of α, N = 10, and ν = λ = 0.5. The present method is compared with the shifted Chebyshev spectral tau (SCT) method given in [48]. As we see from Table 7, it is clear that the result obtained by the present method for each choice of the parameter α is superior to that obtained by SCT method. Fig 3 shows the absolute error function at N = 10, α = 0 and ν = λ = 0.5. The obtained results of this example show that the present method is very accurate by selecting a few number of fractional-order generalized Laguerre generalized functions.


New operational matrices for solving fractional differential equations on the half-line.

Bhrawy AH, Taha TM, Alzahrani EO, Baleanu D, Alzahrani AA - PLoS ONE (2015)

Graph of the absolute error function for N = 10, α = 0 and ν = λ = 0.5, for Example 6.
© Copyright Policy
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4440753&req=5

pone.0126620.g003: Graph of the absolute error function for N = 10, α = 0 and ν = λ = 0.5, for Example 6.
Mentions: In Table 7, we list the results obtained by the fractional-order generalized Laguerre generalized collocation (FGLC) method with various choices of α, N = 10, and ν = λ = 0.5. The present method is compared with the shifted Chebyshev spectral tau (SCT) method given in [48]. As we see from Table 7, it is clear that the result obtained by the present method for each choice of the parameter α is superior to that obtained by SCT method. Fig 3 shows the absolute error function at N = 10, α = 0 and ν = λ = 0.5. The obtained results of this example show that the present method is very accurate by selecting a few number of fractional-order generalized Laguerre generalized functions.

Bottom Line: 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.

View Article: PubMed Central - PubMed

Affiliation: Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia; Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef, Egypt.

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.

No MeSH data available.