A Quadrature Tau Method for Solving Fractional Integro-differential Equations in the Caputo Sense
-
2639
Downloads
-
3772
Views
Authors
A. Yousefi
- Department of Computer Science, Faculty of Mathematical Science and Computer, Kharazmi University, Tehran, Iran
T. Mahdavi Rad
- Department of Mathematics, Arak branch, Islamic Azad University, Arak, Iran.
S. G. Shafiei
- Department of Computer Science, Faculty of Mathematical Science and Computer, Kharazmi University, Tehran, Iran
Abstract
In this article, we develop a direct solution technique for solving fractional integro-differential equations (FIDEs) in the Caputo sense using a quadrature shifted Legendre Tau (Q-SLT) method. The spatial approximation is based on shifted Legendre polynomials. A new formula expressing explicitly any fractional-order derivatives of shifted Legendre polynomials of any degree in terms of shifted Legendre polynomials themselves is proved. Extension of the Tau method for FIDEs is treated using the shifted Legendre–Gauss–Lobatto quadrature. The method is illustrated by considering some examples whose exact solutions are available. The results obtained through this method are stable and comparable with the existing methods for a variety of problems with practical applications.
Share and Cite
ISRP Style
A. Yousefi, T. Mahdavi Rad, S. G. Shafiei, A Quadrature Tau Method for Solving Fractional Integro-differential Equations in the Caputo Sense, Journal of Mathematics and Computer Science, 15 (2015), no. 2, 97-107
AMA Style
Yousefi A., Rad T. Mahdavi, Shafiei S. G., A Quadrature Tau Method for Solving Fractional Integro-differential Equations in the Caputo Sense. J Math Comput SCI-JM. (2015); 15(2):97-107
Chicago/Turabian Style
Yousefi, A., Rad, T. Mahdavi, Shafiei, S. G.. "A Quadrature Tau Method for Solving Fractional Integro-differential Equations in the Caputo Sense." Journal of Mathematics and Computer Science, 15, no. 2 (2015): 97-107
Keywords
- Shifted Legendre Tau method
- Fractional-order derivative
- Caputo derivative
- Fractional Integro-differential.
MSC
References
-
[1]
M. Kot, Elements of Mathematical Ecology, Cambridge University Press, (2001)
-
[2]
D.-B. Pougaza, The Lotka integral equation as a stable population model, Postgraduate Essay, African Institute for Mathematical Sciences (AIMS), (2007)
-
[3]
I. D. Kopeikin, V. P. Shishkin, Integral form of the general solution of equations of steady-state thermo elasticity, J. Appl. Math. Mech. (PMM U.S.S.R.), 48(1) (1984), 117–119.
-
[4]
A. J. Lotka, On an integral equation in population analysis, Ann. Math. Stat , 10 (1939), 144–161.
-
[5]
R. R. Nigmatullin, S. O. Nelson, Recognition of the “fractional” kinetics in complex systems, Dielectric properties of fresh fruits and vegetables from 0.01 to 1.8 GHz, Signal Processing, 86 (2006), 2744-2759.
-
[6]
I. S. Jesus, J. A.T. Machado, J. B. Cunha, Fractional electrical impedances in botanical elements, Journal of Vibration and Control , 14 (2008), 1389-1402.
-
[7]
I. S. Jesus, J. A.T. Machado, J. B. Cunha, Fractional order electrical impedance of fruits and vegetables, in Proceedings of the 25th IASTED International Conference MODELLING, IDENTIFICATION, AND CONTROL, Lanzarote, Canary Islands, Spain (2006)
-
[8]
L. M. Petrovic, D. T. Spasic, T. M. Atanackovic, On a mathematical model of a human root dentin, Dental Materials, 21 (2005), 125-128.
-
[9]
V. D. Djordjevi¢, J. Jari¢, B. Fabry, J. J. Fredberg, D. Stamenovi¢, Fractional derivatives embody essential features of cell rheological behavior, Annals of Biomedical Engineering , 31 (2003), 692-699.
-
[10]
K. S. Cole, Electric conductance of biological systems, in Proc. Cold Spring Harbor Symp. Quant. Biol, Cold Spring Harbor, New York, (1993), 107-116.
-
[11]
T. J. Anastasio, The fractional-order dynamics of bainstem vestibulo-oculomotor neurons, Biological Cybernetics , 72 (1994), 69-79.
-
[12]
M. F. Al-Jamal, E. A. Rawashdeh, The approximate solution of fractional integro-differential equations, Int. J. Contemp. Math. Science, 4 (2009), 1067-1078.
-
[13]
W. G. El-Sayed, A. M. A. El-sayed, On the fractional integral equations of mixed type integro-differential equations of fractional orders, Applied Mathematics and Computation, 154 (2004), 461-467.
-
[14]
S. Esmaeili, M. Shamsi, A pseudo-spectral scheme for the approximate solution of a family of fractional differential equations, Commun. Nonlinear Sci. Numer. Simul. , 16 (2011), 3646–3654.
-
[15]
A. Neamaty, B. Agheli, R. Darzi, Solving fractional partial differential equation by using wavelet operational method, J. Math. Computer Sci. , 7 (2013), 230 – 240.
-
[16]
F. Ghoreishi, S. Yazdani, An extension of the spectral Tau method for numerical solution of multi-order fractional differential equations with convergence analysis, Comput. Math. Appl. , 61 (2011), 30–43.
-
[17]
A. Pedas, E. Tamme, On the convergence of spline collocation methods for solving fractional differential equations, J. Comput. Appl. Math. , 235 (2011), 3502–3514.
-
[18]
S. K. Vanani, A. Aminataei, Tau approximate solution of fractional partial differential equations, Comput. Math. Appl. doi:10.1016/j.camwa.2011.03.013. , (2011),
-
[19]
M. Alipour, D. Rostamy, Solving nonlinear fractional differential equations by Bernstein polynomials operational matrices, J. Math. Computer Sci., 3 (2012), 185-196.
-
[20]
E. H. Doha, A. H. Bhrawy, S. S. Ezz-Eldien, Efficient Chebyshev spectral methods for solving multi-term fractional orders differential equations, Appl. Math. Model. , doi:10.1016/j.apm.2011.05.011. (2011)
-
[21]
E. H. Doha, A. H. Bhrawy, R. M. Hafez, A Jacobi dual-Petrov–Galerkin method for solving some odd-order ordinary differential equations, Abstr. Appl. Anal., doi:10.1155/2011/947230. (2011)
-
[22]
A. H. Bhrawy, A. S. Alofi, S. S. Ezz-Eldien, A quadrature Tau method for fractional differential equations with variable coefficients, App. Math. Letters, 24 (2011), 2146–2152.
-
[23]
J. Shen, T. Tang, Li-Lian Wang, Spectral methods, Algorithms, Analysis and applications, Springer-Verlag Berlin Heidelberg, (2011)