A new Legendre collocation method for solving a two-dimensional fractional diffusion equation. (English) Zbl 1468.65161

Summary: 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.


65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
35R11 Fractional partial differential equations
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
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[1] Canuto, C.; Hussaini, M. Y.; Quarteroni, A.; Zang, T. A., Spectral Methods: Fundamentals in Single Domains, (2006), New York, NY, USA: Springer, New York, NY, USA · Zbl 1093.76002
[2] Schamel, H.; Elsaesser, K., The application of the spectral method to nonlinear wave propagation, Journal of Computational Physics, 22, 4, 501-516, (1976) · Zbl 0344.65055
[3] Abd-Elhameed, W. M.; Doha, E. H.; Youssri, Y. H., Efficient spectral-Petrov-Galerkin methods for third- and fifth-order differential equations using general parameters generalized Jacobi polynomials, Quaestiones Mathematicae, 36, 15-38, (2013) · Zbl 1274.65222
[4] Doha, E. H.; Bhrawy, A. H.; Abdelkawy, M. A.; Hafez, R. M., A Jacobi collocation approximation for nonlinear coupled viscous Burgers’ equation, Central European Journal of Physics, 12, 111-122, (2014)
[5] Adibi, H.; Rismani, A. M., On using a modified Legendre-spectral method for solving singular IVPs of Lane-Emden type, Computers and Mathematics with Applications, 60, 7, 2126-2130, (2010) · Zbl 1205.65201
[6] Doha, E. H.; Bhrawy, A. H.; Abdelkawy, M. A.; van Gorder, R. A., Jacobi-Gauss-Lobatto collocation method for the numerical solution of \(1 + 1\) nonlinear Schrodinger equations, Journal of Computational Physics, 261, 244-255, (2014) · Zbl 1349.65511
[7] Doha, E. H.; Bhrawy, A. H.; Baleanu, D.; Hafez, R. M., A new Jacobi rational-Gauss collocation method for numerical solution of generalized Pantograph equations, Applied Numerical Mathematics, 77, 43-54, (2014) · Zbl 1302.65175
[8] Mahfouz, F. M., Numerical simulation of free convection within an eccentric annulus filled with micropolar fluid using spectral method, Applied Mathematics and Computation, 219, 5397-5409, (2013) · Zbl 1282.76144
[9] Ma, J.; Li, B.-W.; Howell, J. R., 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, 71, 35-43, (2014)
[10] Ma, X.; Huang, C., Spectral collocation method for linear fractional integro-differential equations, Applied Mathematical Modelling, 38, 1434-1448, (2014) · Zbl 1427.65421
[11] Abd-Elhameed, W. M.; Doha, E. H.; Youssri, Y. H., New wavelets collocation method for solving second-order multipoint boundary value problems using Chebyshev polynomials of third and fourth, Abstract and Applied Analysis, 2013, (2013) · Zbl 1291.65238
[12] Lau, S. R.; Price, R. H., Sparse spectral-tau method for the three-dimensional helically reduced wave equation on two-center domains, Journal of Computational Physics, 231, 7695-7714, (2012) · Zbl 1284.65176
[13] Ghoreishi, F.; Yazdani, S., An extension of the spectral Tau method for numerical solution of multi-order fractional differential equations with convergence analysis, Computers and Mathematics with Applications, 61, 1, 30-43, (2011) · Zbl 1207.65108
[14] Doha, E. H.; Bhrawy, A. H., An efficient direct solver for multidimensional elliptic Robin boundary value problems using a Legendre spectral-Galerkin method, Computers and Mathematics with Applications, 64, 558-571, (2012) · Zbl 1252.65194
[15] Boaca, T.; Boaca, I., Spectral galerkin method in the study of mass transfer in laminar and turbulent flows, Computer Aided Chemical Engineering, 24, 99-104, (2007)
[16] Doha, E. H.; Bhrawy, A. H.; Hafez, R. M., A Jacobi-Jacobi dual-Petrov-Galerkin method for third- and fifth-order differential equations, Mathematical and Computer Modelling, 53, 9-10, 1820-1832, (2011) · Zbl 1219.65077
[17] Abd-Elhameed, W. M.; Doha, E. H.; Bassuony, M. A., Two Legendre-Dual-Petrov-Galerkin algorithms for solving the integrated forms of high odd-order boundary value problems, The Scientific World Journal, 2013, (2013) · Zbl 1286.65093
[18] Wang, L.; Ma, Y.; Meng, Z., Haar wavelet method for solving fractional partial differential equations numerically, Applied Mathematics and Computation, 227, 66-76, (2014) · Zbl 1364.65213
[19] Golbabai, A.; Javidi, M., A numerical solution for non-classical parabolic problem based on Chebyshev spectral collocation method, Applied Mathematics and Computation, 190, 1, 179-185, (2007) · Zbl 1122.65390
[20] Bhrawy, A. H., A Jacobi-Gauss-Lobatto collocation method for solving generalized Fitzhugh-Nagumo equation with time-dependent coefficients, Applied Mathematics and Computation, 222, 255-264, (2013) · Zbl 1329.65234
[21] Doha, E. H.; Bhrawy, A. H.; Hafez, R. M., On shifted Jacobi spectral method for high-order multi-point boundary value problems, Communications in Nonlinear Science and Numerical Simulation, 17, 10, 3802-3810, (2012) · Zbl 1251.65112
[22] Garrappa, R.; Popolizio, M., On the use of matrix functions for fractional partial differential equations, Mathematics and Computers in Simulation, 81, 5, 1045-1056, (2011) · Zbl 1210.65162
[23] Pedas, A. A.; Tamme, E., Numerical solution of nonlinear fractional differential equations by spline collocation methods, Journal of Computational and Applied Mathematics, 255, 216-230, (2014) · Zbl 1291.65247
[24] Gao, F.; Lee, X.; Fei, F.; Tong, H.; Deng, Y.; Zhao, H., Identification time-delayed fractional order chaos with functional extrema model via differential evolution, Expert Systems With Applications, 41, 1601-1608, (2014)
[25] Zhao, Y.; Cheng, D. F.; Yang, X. J., Approximation solutions for local fractional Schrödinger equation in the one-dimensional Cantorian system, Advances in Mathematical Physics, 2013, (2013) · Zbl 1292.81050
[26] Bhrawy, A. H.; Al-Shomrani, M. M., A shifted Legendre spectral method for fractional-order multi-point boundary value problems, Advances in Difference Equations, 2012, (2012) · Zbl 1280.65074
[27] Kirchner, J. W.; Feng, X.; Neal, C., Frail chemistry and its implications for contaminant transport in catchments, Nature, 403, 6769, 524-526, (2000)
[28] Pedas, A.; Tamme, E., Piecewise polynomial collocation for linear boundary value problems of fractional differential equations, Journal of Computational and Applied Mathematics, 236, 13, 3349-3359, (2012) · Zbl 1245.65104
[29] Ahmadian, A.; Suleiman, M.; Salahshour, S.; Baleanu, D., A Jacobi operational matrix for solving a fuzzy linear fractional differential equation, Advances in Difference Equations, 2013, (2013) · Zbl 1380.34004
[30] Bhrawy, A. H.; Alghamdi, M. A., A shifted Jacobi-Gauss-Lobatto collocation method for solving nonlinear fractional Langevin equation involving two fractional orders in different intervals, Boundary Value Problems, 2012, (2012) · Zbl 1280.65079
[31] Hilfer, R., Applications of Fractional Calculus in Physics, (2000), Singapore: Word Scientific, Singapore · Zbl 0998.26002
[32] Kotomin, E.; Kuzovkov, V., Modern Aspects of Diffusion-Controlled Reactions: Cooperative Phenomena in Bimolecular Processes. Modern Aspects of Diffusion-Controlled Reactions: Cooperative Phenomena in Bimolecular Processes, Comprehensive Chemical Kinetics, (1996), Elsevier
[33] Baleanu, D.; Diethelm, K.; Scalas, E.; Trujillo, J. J., Fractional Calculus Models and Numerical Methods. Fractional Calculus Models and Numerical Methods, Series on Complexity, Nonlinearity and Chaos, (2012), Singapore: World Scientific, Singapore · Zbl 1248.26011
[34] Cantrell, R. S.; Cosner, C., Spatial Ecology via Reaction Diffusion Equations, (2004), Wiley · Zbl 1058.92041
[35] Wang, H.; Du, N., Fast solution methods for space-fractional diffusion equations, Journal of Computational and Applied Mathematics, 255, 376-383, (2014) · Zbl 1291.65324
[36] Bhrawy, A. H.; Baleanu, D., A spectral Legendre-Gauss-Lobatto collocation method for a space-fractional advection diffusion equations with variable coefficients, Reports on Mathematical Physics, 72, 219-233, (2013) · Zbl 1292.65109
[37] Doha, E. H.; Bhrawy, A. H.; Ezz-Eldien, S. S., Numerical approximations for fractional diffusion equations via a Chebyshev spectral-tau method, Central European Journal of Physics, 11, 1494-1503, (2013)
[38] Liu, F.; Zhuang, P.; Turner, I.; Burrage, K.; Anh, V., A new fractional finite volume method for solving the fractional diffusion equation, Applied Mathematical Modelling, (2013) · Zbl 1429.65213
[39] Yuste, S. B.; Quintana-Murillo, J., A finite difference method with non-uniform timesteps for fractional diffusion equations, Computer Physics Communications, 183, 2594-2600, (2012) · Zbl 1268.65120
[40] Wang, K.; Wang, H., A fast characteristic finite difference method for fractional advection-diffusion equations, Advances in Water Resources, 34, 7, 810-816, (2011)
[41] Miller, K.; Ross, B., An Introduction to the Fractional Calaulus and Fractional Differential Equations, (1993), New York, NY, USA: John Wiley & Sons, New York, NY, USA
[42] Podluny, I., Fractional Differential Equations, (1999), San Diego, Calif, USA: Academic Press, San Diego, Calif, USA
[43] Tadjeran, C.; Meerschaert, M. M., A second-order accurate numerical method for the two-dimensional fractional diffusion equation, Journal of Computational Physics, 220, 2, 813-823, (2007) · Zbl 1113.65124
[44] Baleanu, D.; Bhrawy, A. H.; Taha, T. M., Two efficient generalized Laguerre spectral algorithms for fractional initial value problems, Abstract and Applied Analysis, 2013, (2013) · Zbl 1291.65240
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