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Research Papers

An Efficient Legendre Spectral Tau Matrix Formulation for Solving Fractional Subdiffusion and Reaction Subdiffusion Equations

[+] Author and Article Information
E. H. Doha

Department of Mathematics,
Faculty of Science,
Cairo University,
Giza 12613, Egypt
e-mail: eiddoha@frcu.eun.eg

A. H. Bhrawy

Department of Mathematics,
Faculty of Science,
King Abdulaziz University,
Jeddah 21589, Saudi Arabia;
Department of Mathematics,
Faculty of Science,
Beni-Suef University,
Beni-Suef 62511, Egypt
e-mail: alibhrawy@yahoo.co.uk

S. S. Ezz-Eldien

Department of Basic Science,
Institute of Information Technology,
Modern Academy,
Cairo 11731, Egypt
e-mail: s_sezeldien@yahoo.com

Manuscript received January 26, 2014; final manuscript received June 19, 2014; published online January 14, 2015. Assoc. Editor: J. A. Tenreiro Machado.

J. Comput. Nonlinear Dynam 10(2), 021019 (Mar 01, 2015) (8 pages) Paper No: CND-14-1030; doi: 10.1115/1.4027944 History: Received January 26, 2014; Revised June 19, 2014; Online January 14, 2015

In this work, we discuss an operational matrix approach for introducing an approximate solution of the fractional subdiffusion equation (FSDE) with both Dirichlet boundary conditions (DBCs) and Neumann boundary conditions (NBCs). We propose a spectral method in both temporal and spatial discretizations for this equation. Our approach is based on the space-time shifted Legendre tau-spectral method combined with the operational matrix of fractional integrals, described in the Riemann–Liouville sense. The main characteristic behind this approach is to reduce such problems to those of solving systems of algebraic equations in the unknown expansion coefficients of the sought-for spectral approximations. In addition, this approach is also investigated for solving the FSDE with the variable coefficients and the fractional reaction subdiffusion equation (FRSDE). For conforming the validity and accuracy of the numerical scheme proposed, four numerical examples with their approximate solutions are presented. Also, comparisons between our numerical results and those obtained by compact finite difference method (CFDM), Box-type scheme (B-TS), and FDM with Fourier analysis (FA) are introduced.

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References

Couceiro, M. S., Ferreira, N. M. F., and Machado, J. A. T., 2010, “Application of Fractional Algorithms in the Control of a Robotic Bird,” Commun. Nonlinear Sci. Numer. Simul., 15(4), pp. 895–910. [CrossRef]
Jesus, I. S., and Machado, J. A. T., “Fractional Control With a Smith Predictor,” ASME J. Comput. Nonlinear Dyn., 6(3), p. 031014. [CrossRef]
Kilbas, A. A., Srivastava, H. M., and Trujillo, J. J., 2006, Theory and Applications of Fractional Differential Equations (North-Holland Mathematics Studies, Vol. 204), Elsevier Science B.V., Amsterdam.
Machado, J. A. T., 2012, “The Effect of Fractional Order in Variable Structure Control,” Comput. Math. Appl., 64(10), pp. 3340–3350. [CrossRef]
Machado, J. A. T., Kiryakova, V., and Mainardi, F., 2011, “Recent History of Fractional Calculus,” Commun. Nonlinear Sci. Numer. Simul., 16(3), pp. 1140–1153. [CrossRef]
Mainardi, F., 1997, “Fractional Calculus: Some Basic Problems in Continuum and Statistical Mechanics,” Fractals and Fractional Calculus in Continuum Mechanics, A.Carpinteri, and F.Mainardi, eds., Springer-Verlag, New York, pp. 291–348.
Miller, K. S., and Ross, B., 1993, An Introduction to The Fractional Calculus and Fractional Differential Equations, Wiley, New York.
Oldham, K. B., and Spanier, J., 1974, The Fractional Calculus, Academic Press, New York.
Podlubny, I., 1999, Fractional Differential Equations (Mathematics in Science and Engineering), Academic Press, New York.
Y.Rossikhin, and M.Shitikova, 2010, “Application of Fractional Calculus for Dynamic Problems of Solid Mechanics: Novel Trends and Recent Results,” ASME Appl. Mech. Rev., 63(1), pp. 1–52. [CrossRef]
Afrouzi, G. A., Talarposhti, R. A., and Ahangar, H., 2012, “Explicit Analytical Solution for a Kind of Time-Fractional Evolution Equations by He's Homotopy Perturbation Methods,” J. Math. Comput. Sci., 4, pp. 278–282.
Odibat, Z., Momani, S., and Xu, H., 2010, “A Reliable Algorithm of Homotopy Analysis Method for Solving Nonlinear Fractional Differential Equations,” Appl. Math. Modell., 34(3), pp. 593–600. [CrossRef]
Song, L., and Wang, W., 2013, “A New Improved Adomian Decomposition Method and Its Application to Fractional Differential Equations,” Appl. Math. Modell., 37(3), pp. 1590–1598. [CrossRef]
Ganjiani, M., 2010, “Solution of Nonlinear Fractional Differential Equations Using Homotopy Analysis Method,” Appl. Math. Modell., 34(6), pp. 1634–1641. [CrossRef]
Zurigat, M., Momani, S., Odibat, Z., and Alawneh, A., 2010, “The Homotopy Analysis Method for Handling Systems of Fractional Differential Equations,” Appl. Math. Modell., 34(1), pp. 24–35. [CrossRef]
Darzi, R., Mohammadzade, B., Mousavi, S., and Beheshti, R., 2013, “Sumudu Transform Method for Solving Fractional Differential Equations and Fractional Diffusion-Wave Equation,” J. Math. Comput. Sci., 6, pp. 79–84.
Neamaty, A., Agheli, B., and Darzi, R., 2013, “Solving Fractional Partial Differential Equation by Using Wavelet Operational Method,” J. Math. Comput. Sci., 7, pp. 230–240.
Erjaee, G. H., Taghvafard, H., and Alnasr, M., 2011, “Numerical Solution of the High Thermal Loss Problem Presented by a Fractional Differential Equation,” Commun. Nonlinear Sci. Numer. Simul., 16(3), pp. 1356–1362. [CrossRef]
Garrappa, R., 2009, “On Some Explicit Adams Multistep Methods for Fractional Differential Equations,” J. Comput. Appl. Math., 229(2), pp. 392–399. [CrossRef]
Garrappa, R., and Popolizio, M., 2011, “On Accurate Product Integration Rules for Linear Fractional Differential Equations,” J. Comput. Appl. Math., 235(5), pp. 1085–1097. [CrossRef]
Bhrawy, A. H., Tharwat, M. M., and Yildirim, A., 2012, “A New Formula for Fractional Integrals of Chebyshev Polynomials: Application for Solving Multi-Term Fractional Differential Equations,” Appl. Math. Modell., 37(6), pp. 4245–4252. [CrossRef]
Doha, E. H., Bhrawy, A. H., and Ezz-Eldien, S. S., 2011, “Efficient Chebyshev Spectral Methods for Solving Multi-Term Fractional Orders Differential Equations,” Appl. Math. Modell., 35(12), pp. 5662–5672. [CrossRef]
Bhrawy, A. H., Alofi, A. S., and Ezz-Eldien, S. S., 2011, “A Quadrature Tau Method for Variable Coefficients Fractional Differential Equations,” Appl. Math. Lett., 24(12), pp. 2146–2152. [CrossRef]
Bhrawy, A. H., and Al-Shomrani, M. M., 2012, “A Shifted Legendre Spectral Method for Fractional-Order Multi-Point Boundary Value Problems,” Adv. Differ. Equations, 2012, p. 8. [CrossRef]
Doha, E. H., Bhrawy, A. H., Baleanu, D., and Ezz-Eldien, S. S., 2013, “On Shifted Jacobi Spectral Approximations for Solving Fractional Differential Equations,” Appl. Math. Comput., 219(15), pp. 8042–8056. [CrossRef]
Doha, E. H., Bhrawy, A. H., and Ezz-Eldien, S. S., 2011, “A Chebyshev Spectral Method Based on Operational Matrix for Initial and Boundary Value Problems of Fractional Order,” Comput. Math. Appl., 62(5), pp. 2364–2373. [CrossRef]
Doha, E. H., Bhrawy, A. H., and Ezz-Eldien, S. S., 2012, “A New Jacobi Operational Matrix: An Application for Solving Fractional Differential Equations,” Appl. Math. Modell., 36(10), pp. 4931–4943. [CrossRef]
Karimi Vanani, S., and Aminataei, A., 2011, “Operational Tau Approximation for a General Class of Fractional Integro-Differential Equations,” Comput. Appl. Math., 30(3), pp. 655–674. [CrossRef]
Mokhtary, P., and Ghoreishi, F., 2011, “The L2–Convergence of the Legendre Spectral Tau Matrix Formulation for Nonlinear Fractional Integro Differential Equations,” Numer. Algorithms, 58(4), pp. 475–496. [CrossRef]
Saadatmandi, A., and Dehghan, M., 2010, “A New Operational Matrix for Solving Fractional-Order Differential Equations,” Comput. Math. Appl., 59(3), pp. 1326–1336. [CrossRef]
Bhrawy, A. H., Alghamdi, M. A., and Taha, T. M., 2012, “A New Modified Generalized Laguerre Operational Matrix of Fractional Integration for Solving Fractional Differential Equations on the Half Line,” Adv. Differ. Equations, 2012, p. 179. [CrossRef]
Bhrawy, A. H., and Alofi, A. S., 2013, “The Operational Matrix of Fractional Integration for Shifted Chebyshev Polynomials,” Appl. Math. Lett., 26(1), pp. 25–31. [CrossRef]
Akrami, M. H., Atabakzadeh, M. H., and Erjaee, G. H., 2013, “The Operational Matrix of Fractional Integration for Shifted Legendre Polynomials,” Iran. J. Sci. Technol., 37, pp. 439–444.
Doha, E. H., Bhrawy, A. H., and Ezz-Eldien, S. S., 2013, “Numerical Approximations for Fractional Diffusion Equations via a Chebyshev Spectral-Tau Method,” Cent. Eur. J. Phys., 11(10), pp. 1494–1503. [CrossRef]
Ren, R., Li, H., Jiang, W., and Song, M., 2013, “An Efficient Chebyshev-Tau Method for Solving the Space Fractional Diffusion Equations,” Appl. Math. Comput., 224, pp. 259–267. [CrossRef]
Saadatmandi, A., and Dehghan, M., 2011, “A Tau Approach for Solution of the Space Fractional Diffusion Equation,” Comput. Math. Appl., 62(3), pp. 1135–1142. [CrossRef]
Bhrawy, A. H., Doha, E. H., Baleanu, D., and Ezz-Eldien, S. S., “A Spectral Tau Algorithm Based on Jacobi Operational Matrix for Numerical Solution of Time Fractional Diffusion-Wave Equations,” J. Comput. Phys. (in press).
Gao, G., and Sun, Z., 2011, “A Compact Finite Difference Scheme for the Fractional Sub-Diffusion Equations,” J. Comput. Phys., 230(3), pp. 586–595. [CrossRef]
Zhang, Y. N., and Sun, Z. Z., 2011, “Alternating Direction Implicit Schemes for the Two-Dimensional Sub-Diffusion Equation,” J. Comput. Phys., 230(24), pp. 8713–8728. [CrossRef]
Cui, M. R., 2012, “Compact Alternating Direction Implicit Method for Two-Dimensional Time Fractional Diffusion Equation,” J. Comput. Phys., 231(6), pp. 2621–2633. [CrossRef]
Chen, C., Liu, F., Turner, I., and Anh, V., 2010, “Numerical Schemes and Multivariate Extrapolation of a Two-Dimensional Anomalous Sub-Diffusion Equation,” Numer. Algorithms, 54(1), pp. 1–21. [CrossRef]
Langlands, T. A. M., and Henry, B. I., 2005, “The Accuracy and Stability of an Implicit Solution Method for the Fractional Diffusion Equation,” J. Comput. Phys., 205(2), pp. 719–736. [CrossRef]
Povstenko, Y., 2010, “Signaling Problem for Time-Fractional Diffusion-Wave Equation in a Half-Space in the Case of Angular Symmetry,” Nonlinear Dyn., 59(4), pp. 593–605. [CrossRef]
Zhao, X., and Sun, Z., 2011, “A Box-Type Scheme for Fractional Sub-Diffusion Equation With Neumann Boundary Conditions,” J. Comput. Phys., 230(15), pp. 6061–6074. [CrossRef]
Ren, J., Sun, Z., and Zhao, X., 2013, “Compact Difference Scheme for the Fractional Sub-Diffusion Equation With Neumann Boundary Conditions,” J. Comput. Phys., 232(1), pp. 456–467. [CrossRef]
Cui, M., 2009, “Compact Finite Difference Method for the Fractional Diffusion Equation,” J. Comput. Phys., 228(20), pp. 7792–7804. [CrossRef]
Zhao, X., and Xu, Q., “Efficient Numerical Schemes for Fractional Sub-Diffusion Equation With the Spatially Variable Coefficient,” Appl. Math. Modell., (in press).
Chen, C., Liu, F., and Burrage, K., 2008, “Finite Difference Methods and a Fourier Analysis for the Fractional Reaction-Subdiffusion Equation,” Appl. Math. Comput., 198(2), pp. 754–769. [CrossRef]
Canuto, C., Hussaini, M. Y., Quarteroni, A., and Zang, T. A., 1989, Spectral Methods in Fluid Dynamics, Springer-Verlag, New York.

Figures

Grahic Jump Location
Fig. 1

Error function at n = m = 12 and γ = 0.25 for example 1

Grahic Jump Location
Fig. 2

Error function at n = m = 12 and γ = 0.75 for example 1

Grahic Jump Location
Fig. 3

Error function at n = m = 16 and γ = 0.30 for example 2

Grahic Jump Location
Fig. 4

Error function at n = m = 16 and γ = 0.50 for example 2

Grahic Jump Location
Fig. 5

Error function at n = m = 16 and γ = 0.70 for example 2

Grahic Jump Location
Fig. 6

Error function at n = m = 16 and γ = 0.90 for example 2

Grahic Jump Location
Fig. 7

Error function at n = m = 16 and γ = 0.50 for example 3

Grahic Jump Location
Fig. 8

Error function at n = m = 16 and γ = 0.75 for example 3

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