## Abstract and Applied Analysis

- Abstr. Appl. Anal.
- Volume 2014, Special Issue (2014), Article ID 636191, 10 pages.

### A New Legendre Collocation Method for Solving a Two-Dimensional Fractional Diffusion Equation

**Full-text: Access denied (no subscription detected) **

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

#### Abstract

A new spectral shifted Legendre Gauss-Lobatto collocation (SL-GL-C) method is developed and analyzed to solve a class of two-dimensional initial-boundary fractional diffusion equations with variable coefficients. The method depends basically on the fact that an expansion in a series of shifted Legendre polynomials ${P}_{L,n}(x){P}_{L,m}(y)$, for the function and its space-fractional derivatives occurring in the partial fractional differential equation (PFDE), is assumed; the expansion coefficients are then determined by reducing the PFDE with its boundary and initial conditions to a system of ordinary differential equations (SODEs) for these coefficients. This system may be solved numerically by using the fourth-order implicit Runge-Kutta (IRK) method. This method, in contrast to common finite-difference and finite-element methods, has the exponential rate of convergence for the two spatial discretizations. Numerical examples are presented in the form of tables and graphs to make comparisons with the results obtained by other methods and with the exact solutions more easier.

#### Article information

**Source**

Abstr. Appl. Anal., Volume 2014, Special Issue (2014), Article ID 636191, 10 pages.

**Dates**

First available in Project Euclid: 6 October 2014

**Permanent link to this document**

https://projecteuclid.org/euclid.aaa/1412607137

**Digital Object Identifier**

doi:10.1155/2014/636191

**Mathematical Reviews number (MathSciNet)**

MR3208554

**Zentralblatt MATH identifier**

07022794

#### Citation

Bhrawy, A. H. A New Legendre Collocation Method for Solving a Two-Dimensional Fractional Diffusion Equation. Abstr. Appl. Anal. 2014, Special Issue (2014), Article ID 636191, 10 pages. doi:10.1155/2014/636191. https://projecteuclid.org/euclid.aaa/1412607137

#### References

- C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang,
*Spectral Methods: Fundamentals in Single Domains*, Springer, New York, NY, USA, 2006.Mathematical Reviews (MathSciNet): MR2223552 - H. Schamel and K. Elsaesser, “The application of the spectral method to nonlinear wave propagation,”
*Journal of Computational Physics*, vol. 22, no. 4, pp. 501–516, 1976.Zentralblatt MATH: 0344.65055

Mathematical Reviews (MathSciNet): MR449164

Digital Object Identifier: doi:10.1016/0021-9991(76)90046-2 - W. M. Abd-Elhameed, E. H. Doha, and Y. H. Youssri, “Efficient spectral-Petrov-Galerkin methods for third- and fifth-order differential equations using general parameters generalized Jacobi polynomials,”
*Quaestiones Mathematicae*, vol. 36, pp. 15–38, 2013.Zentralblatt MATH: 1274.65222

Mathematical Reviews (MathSciNet): MR3043668

Digital Object Identifier: doi:10.2989/16073606.2013.779945 - E. H. Doha, A. H. Bhrawy, M. A. Abdelkawy, and R. M. Hafez, “A Jacobi collocation approximation for nonlinear coupled viscous Burgers' equation,”
*Central European Journal of Physics*, vol. 12, pp. 111–122, 2014. - H. Adibi and A. M. Rismani, “On using a modified Legendre-spectral method for solving singular IVPs of Lane-Emden type,”
*Computers and Mathematics with Applications*, vol. 60, no. 7, pp. 2126–2130, 2010. - E. H. Doha, A. H. Bhrawy, M. A. Abdelkawy, and R. A. van Gorder, “Jacobi-Gauss-Lobatto collocation method for the numerical solution of $1+1$ nonlinear Schrodinger equations,”
*Journal of Computational Physics*, vol. 261, pp. 244–255, 2014.Mathematical Reviews (MathSciNet): MR3161196

Digital Object Identifier: doi:10.1016/j.jcp.2014.01.003 - E. H. Doha, A. H. Bhrawy, D. Baleanu, and R. M. Hafez, “A new Jacobi rational-Gauss collocation method for numerical solution of generalized Pantograph equations,”
*Applied Numerical Mathematics*, vol. 77, pp. 43–54, 2014.Mathematical Reviews (MathSciNet): MR3145364

Digital Object Identifier: doi:10.1016/j.apnum.2013.11.003 - F. M. Mahfouz, “Numerical simulation of free convection within an eccentric annulus filled with micropolar fluid using spectral method,”
*Applied Mathematics and Computation*, vol. 219, pp. 5397–5409, 2013.Zentralblatt MATH: 1282.76144

Mathematical Reviews (MathSciNet): MR3009497

Digital Object Identifier: doi:10.1016/j.amc.2012.11.038 - J. Ma, B.-W. Li, and J. R. Howell, “Thermal radiation heat transfer in one- and two-dimensional enclosures using the spectral collocation method with full spectrum k-distribution model,”
*International Journal of Heat and Mass Transfer*, vol. 71, pp. 35–43, 2014. - X. Ma and C. Huang, “Spectral collocation method for linear fractional integro-differential equations,”
*Applied Mathematical Modelling*, vol. 38, pp. 1434–1448, 2014.Mathematical Reviews (MathSciNet): MR3164089

Digital Object Identifier: doi:10.1016/j.apm.2013.08.013 - W. M. Abd-Elhameed, E. H. Doha, and Y. H. Youssri, “New wavelets collocation method for solving second-order multipoint boundary value problems using Chebyshev polynomials of third and fourth,”
*Abstract and Applied Analysis*, vol. 2013, Article ID 542839, 9 pages, 2013.Mathematical Reviews (MathSciNet): MR3121515 - S. R. Lau and R. H. Price, “Sparse spectral-tau method for the three-dimensional helically reduced wave equation on two-center domains,”
*Journal of Computational Physics*, vol. 231, pp. 7695–7714, 2012.Mathematical Reviews (MathSciNet): MR2972854

Digital Object Identifier: doi:10.1016/j.jcp.2012.07.006 - F. Ghoreishi and S. Yazdani, “An extension of the spectral Tau method for numerical solution of multi-order fractional differential equations with convergence analysis,”
*Computers and Mathematics with Applications*, vol. 61, no. 1, pp. 30–43, 2011. - E. H. Doha and A. H. Bhrawy, “An efficient direct solver for multidimensional elliptic Robin boundary value problems using a Legendre spectral-Galerkin method,”
*Computers and Mathematics with Applications*, vol. 64, pp. 558–571, 2012. - T. Boaca and I. Boaca, “Spectral galerkin method in the study of mass transfer in laminar and turbulent flows,”
*Computer Aided Chemical Engineering*, vol. 24, pp. 99–104, 2007. - E. H. Doha, A. H. Bhrawy, and R. M. Hafez, “A Jacobi-Jacobi dual-Petrov-Galerkin method for third- and fifth-order differential equations,”
*Mathematical and Computer Modelling*, vol. 53, no. 9-10, pp. 1820–1832, 2011.Zentralblatt MATH: 1219.65077

Mathematical Reviews (MathSciNet): MR2782868

Digital Object Identifier: doi:10.1016/j.mcm.2011.01.002 - W. M. Abd-Elhameed, E. H. Doha, and M. A. Bassuony, “Two Legendre-Dual-Petrov-Galerkin algorithms for solving the integrated forms of high odd-order boundary value problems,”
*The Scientific World Journal*, vol. 2013, Article ID 309264, 11 pages, 2013. - L. Wang, Y. Ma, and Z. Meng, “Haar wavelet method for solving fractional partial differential equations numerically,”
*Applied Mathematics and Computation*, vol. 227, pp. 66–76, 2014.Mathematical Reviews (MathSciNet): MR3146297

Digital Object Identifier: doi:10.1016/j.amc.2013.11.004 - A. Golbabai and M. Javidi, “A numerical solution for non-classical parabolic problem based on Chebyshev spectral collocation method,”
*Applied Mathematics and Computation*, vol. 190, no. 1, pp. 179–185, 2007.Zentralblatt MATH: 1122.65390

Mathematical Reviews (MathSciNet): MR2335438

Digital Object Identifier: doi:10.1016/j.amc.2007.01.033 - A. H. Bhrawy, “A Jacobi-Gauss-Lobatto collocation method for solving generalized Fitzhugh-Nagumo equation with time-dependent coefficients,”
*Applied Mathematics and Computation*, vol. 222, pp. 255–264, 2013.Mathematical Reviews (MathSciNet): MR3115866

Digital Object Identifier: doi:10.1016/j.amc.2013.07.056 - E. H. Doha, A. H. Bhrawy, and R. M. Hafez, “On shifted Jacobi spectral method for high-order multi-point boundary value problems,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 17, no. 10, pp. 3802–3810, 2012.Zentralblatt MATH: 1251.65112

Mathematical Reviews (MathSciNet): MR2916435

Digital Object Identifier: doi:10.1016/j.cnsns.2012.02.027 - R. Garrappa and M. Popolizio, “On the use of matrix functions for fractional partial differential equations,”
*Mathematics and Computers in Simulation*, vol. 81, no. 5, pp. 1045–1056, 2011.Zentralblatt MATH: 1210.65162

Mathematical Reviews (MathSciNet): MR2769818

Digital Object Identifier: doi:10.1016/j.matcom.2010.10.009 - A. A. Pedas and E. Tamme, “Numerical solution of nonlinear fractional differential equations by spline collocation methods,”
*Journal of Computational and Applied Mathematics*, vol. 255, pp. 216–230, 2014.Mathematical Reviews (MathSciNet): MR3093417

Digital Object Identifier: doi:10.1016/j.cam.2013.04.049 - F. Gao, X. Lee, F. Fei, H. Tong, Y. Deng, and H. Zhao, “Identification time-delayed fractional order chaos with functional extrema model via differential evolution,”
*Expert Systems With Applications*, vol. 41, pp. 1601–1608, 2014. - Y. Zhao, D. F. Cheng, and X. J. Yang, “Approximation solutions for local fractional Schrödinger equation in the one-dimensional Cantorian system,”
*Advances in Mathematical Physics*, vol. 2013, Article ID 291386, 5 pages, 2013. - A. H. Bhrawy and M. M. Al-Shomrani, “A shifted Legendre spectral method for fractional-order multi-point boundary value problems,”
*Advances in Difference Equations*, vol. 2012, article 8, 2012.Zentralblatt MATH: 1280.65074 - J. W. Kirchner, X. Feng, and C. Neal, “Frail chemistry and its implications for contaminant transport in catchments,”
*Nature*, vol. 403, no. 6769, pp. 524–526, 2000. - A. Pedas and E. Tamme, “Piecewise polynomial collocation for linear boundary value problems of fractional differential equations,”
*Journal of Computational and Applied Mathematics*, vol. 236, no. 13, pp. 3349–3359, 2012.Zentralblatt MATH: 1245.65104

Mathematical Reviews (MathSciNet): MR2912696

Digital Object Identifier: doi:10.1016/j.cam.2012.03.002 - A. Ahmadian, M. Suleiman, S. Salahshour, and D. Baleanu, “A Jacobi operational matrix for solving a fuzzy linear fractional differential equation,”
*Advances in Difference Equations*, vol. 2013, article 104, 29 pages, 2013. - A. H. Bhrawy and M. A. Alghamdi, “A shifted Jacobi-Gauss-Lobatto collocation method for solving nonlinear fractional Langevin equation involving two fractional orders in different intervals,”
*Boundary Value Problems*, vol. 2012, article 62, 2012.Zentralblatt MATH: 1280.65079 - R. Hilfer,
*Applications of Fractional Calculus in Physics*, Word Scientific, Singapore, 2000.Mathematical Reviews (MathSciNet): MR1890104 - E. Kotomin and V. Kuzovkov,
*Modern Aspects of Diffusion-Controlled Reactions: Cooperative Phenomena in Bimolecular Processes*, Comprehensive Chemical Kinetics, Elsevier, 1996. - D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo,
*Fractional Calculus Models and Numerical Methods*, Series on Complexity, Nonlinearity and Chaos, World Scientific, Singapore, 2012.Mathematical Reviews (MathSciNet): MR2894576 - R. S. Cantrell and C. Cosner,
*Spatial Ecology via Reaction Diffusion Equations*, Wiley, 2004.Mathematical Reviews (MathSciNet): MR2191264 - H. Wang and N. Du, “Fast solution methods for space-fractional diffusion equations,”
*Journal of Computational and Applied Mathematics*, vol. 255, pp. 376–383, 2014.Mathematical Reviews (MathSciNet): MR3093429

Digital Object Identifier: doi:10.1016/j.cam.2013.06.002 - A. H. Bhrawy and D. Baleanu, “A spectral Legendre-Gauss-Lobatto collocation method for a space-fractional advection diffusion equations with variable coefficients,”
*Reports on Mathematical Physics*, vol. 72, pp. 219–233, 2013.Mathematical Reviews (MathSciNet): MR3164053

Digital Object Identifier: doi:10.1016/S0034-4877(14)60015-X - E. H. Doha, A. H. Bhrawy, and S. S. Ezz-Eldien, “Numerical approximations for fractional diffusion equations via a Chebyshev spectral-tau method,”
*Central European Journal of Physics*, vol. 11, pp. 1494–1503, 2013. - F. Liu, P. Zhuang, I. Turner, K. Burrage, and V. Anh, “A new fractional finite volume method for solving the fractional diffusion equation,”
*Applied Mathematical Modelling*, 2013.Mathematical Reviews (MathSciNet): MR3233813

Digital Object Identifier: doi:10.1016/j.apm.2013.10.007 - S. B. Yuste and J. Quintana-Murillo, “A finite difference method with non-uniform timesteps for fractional diffusion equations,”
*Computer Physics Communications*, vol. 183, pp. 2594–2600, 2012.Zentralblatt MATH: 1268.65120

Mathematical Reviews (MathSciNet): MR2970362

Digital Object Identifier: doi:10.1016/j.cpc.2012.07.011 - K. Wang and H. Wang, “A fast characteristic finite difference method for fractional advection-diffusion equations,”
*Advances in Water Resources*, vol. 34, no. 7, pp. 810–816, 2011. - K. Miller and B. Ross,
*An Introduction to the Fractional Calaulus and Fractional Differential Equations*, John Wiley & Sons, New York, NY, USA, 1993.Mathematical Reviews (MathSciNet): MR1219954 - I. Podluny,
*Fractional Differential Equations*, Academic Press, San Diego, Calif, USA, 1999.Mathematical Reviews (MathSciNet): MR1658022 - C. Tadjeran and M. M. Meerschaert, “A second-order accurate numerical method for the two-dimensional fractional diffusion equation,”
*Journal of Computational Physics*, vol. 220, no. 2, pp. 813–823, 2007.Zentralblatt MATH: 1113.65124

Mathematical Reviews (MathSciNet): MR2284325

Digital Object Identifier: doi:10.1016/j.jcp.2006.05.030 - D. Baleanu, A. H. Bhrawy, and T. M. Taha, “Two efficient generalized Laguerre spectral algorithms for fractional initial value problems,”
*Abstract and Applied Analysis*, vol. 2013, Article ID 546502, 10 pages, 2013. \endinputMathematical Reviews (MathSciNet): MR3066294

### More like this

- Approximate Solutions of Fisher's Type Equations with Variable Coefficients

Bhrawy, A. H. and Alghamdi, M. A., Abstract and Applied Analysis, 2013 - Numerical Solution of a Class of Functional-Differential Equations Using Jacobi Pseudospectral Method

Bhrawy, A. H., Alghamdi, M. A., and Baleanu, D., Abstract and Applied Analysis, 2013 - Fast Spectral Collocation Method for Solving Nonlinear Time-Delayed Burgers-Type Equations with Positive Power Terms

Bhrawy, A. H., Assas, L. M., and Alghamdi, M. A., Abstract and Applied Analysis, 2013

- Approximate Solutions of Fisher's Type Equations with Variable Coefficients

Bhrawy, A. H. and Alghamdi, M. A., Abstract and Applied Analysis, 2013 - Numerical Solution of a Class of Functional-Differential Equations Using Jacobi Pseudospectral Method

Bhrawy, A. H., Alghamdi, M. A., and Baleanu, D., Abstract and Applied Analysis, 2013 - Fast Spectral Collocation Method for Solving Nonlinear Time-Delayed Burgers-Type Equations with Positive Power Terms

Bhrawy, A. H., Assas, L. M., and Alghamdi, M. A., Abstract and Applied Analysis, 2013 - An Efficient Numerical Approach for Solving Nonlinear Coupled Hyperbolic Partial Differential Equations with Nonlocal Conditions

Bhrawy, A. H., Alghamdi, M. A., and Alaidarous, Eman S., Abstract and Applied Analysis, 2014 - New Ultraspherical Wavelets Spectral Solutions for Fractional Riccati Differential Equations

Abd-Elhameed, W. M. and Youssri, Y. H., Abstract and Applied Analysis, 2014 - Numerical Solution of the Fractional Partial Differential Equations by the Two-Dimensional Fractional-Order Legendre Functions

Yin, Fukang, Song, Junqiang, Wu, Yongwen, and Zhang, Lilun, Abstract and Applied Analysis, 2013 - Wavelet Collocation Method for Solving Multiorder Fractional Differential
Equations

Heydari, M. H., Hooshmandasl, M. R., Maalek Ghaini, F. M., and Mohammadi, F., Journal of Applied Mathematics, 2012 - Nonlinear Stability and Convergence of Two-Step Runge-Kutta Methods for Volterra Delay Integro-Differential Equations

Yuan, Haiyan and Song, Cheng, Abstract and Applied Analysis, 2013 - New Spectral Second Kind Chebyshev Wavelets Algorithm for Solving Linear
and Nonlinear Second-Order Differential Equations Involving Singular and
Bratu Type Equations

Abd-Elhameed, W. M., Doha, E. H., and Youssri, Y. H., Abstract and Applied Analysis, 2013 - Stability Analysis of Additive Runge-Kutta Methods for Delay-Integro-Differential Equations

Qin, Hongyu, Wang, Zhiyong, Zhu, Fumin, and Wen, Jinming, International Journal of Differential Equations, 2018