A spectral Legendre-Gauss-Lobatto collocation method for a space-fractional advection diffusion equations with variable coefficients. (English) Zbl 1292.65109

Bhrawy, A. H.; Baleanu, D.

Rep. Math. Phys. 72, No. 2, 219-233 (2013).

Summary: An efficient Legendre-Gauss-Lobatto collocation method is applied to solve the space-fractional advection diffusion equation with nonhomogeneous initial-boundary conditions. The Legendre-Gauss-Lobatto points are used as collocation nodes for spatial fractional derivatives as well as the Caputo fractional derivative. This approach is reducing the problem to the solution of a system of ordinary differential equations in time which can be solved by using any standard numerical techniques. The proposed numerical solutions when compared with the exact solutions reveal that the obtained solution produces highly accurate results. The results show that the proposed method has high accuracy and is efficient for solving the space-fractional advection diffusion equation.

MSC:

65M70	Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
35R11	Fractional partial differential equations
35K57	Reaction-diffusion equations
65M20	Method of lines for initial value and initial-boundary value problems involving PDEs
65L06	Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations

Keywords:

space-fractional advection diffusion equation; spectral method; Legendre-Gauss-Lobatto quadrature; implicit Runge-Kutta method; numerical examples; collocation

References:

[1]	Baleanu, D.; Diethelm, K.; Scalas, E.; Trujillo, J. J., Fractional calculus models and numerical methods, (Series on Complexity, Nonlinearity and Chaos (2012), World Scientific: World Scientific Singapore) · Zbl 1248.26011
[2]	Mainardi, F., Fractional calculus: some basic problems in continuum and statistical mechanics, in:, (Carpinteri, A.; Mainardi, F., Fractals and Fractional Calculus in Continuum Mechanics (1997), Springer: Springer New York) · Zbl 0917.73004
[3]	Spasic, A. M.; Lazarevic, M. P., J. Colloid Interface Sci., 282, 223 (2005)
[4]	Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J., (Theory and Applications of Fractional Differential Equations (2006), Elsevier: Elsevier San Diego) · Zbl 1092.45003
[5]	Bhrawy, A. H.; Alghamdi, M. A., Bound. Value. Prob., 2012, 62 (2012)
[6]	Doha, E. H.; Bhrawy, A. H.; Ezz-Eldien, S. S., Appl. Math. Modell., 35, 5662 (2011)
[7]	Li, X.; Xu, C., SIAM J. Numer. Anal., 45, 2108 (2009)
[8]	Saadatmandi, A.; Dehghan, M., Comput. Math. Appl., 62, 1135 (2011)
[9]	Saadatmandi, A.; Dehghan, M.; Azizi, M. R., Commun. Nonlinear Sci. Numer. Simulat., 17, 4125 (2012)
[10]	(Hilfer, R., Applications of Fractional Calculus in Physics (2000), World Scientific: World Scientific Singapore) · Zbl 0998.26002
[11]	Celik, C.; Duman, M., J. Comput. Phys., 231, 1743 (2012)
[12]	Chen, J.; Liu, F.; Turner, I.; Anh, V., ANZIAM J., 50, 45 (2008)
[13]	Choi, H. W.; Chung, S. K.; Lee, Y. J., Bull. Korean Math. Soc., 47, 1225 (2010)
[14]	Liu, F.; Zhuang, P.; Burrage, K., Comput. Math. Appl., 64, 2990 (2012)
[15]	Canuto, C.; Hussaini, M. Y.; Quarteroni, A.; Zang, T. A., (Spectral Methods: Fundamentals in Single Domains (2006), Springer: Springer New York) · Zbl 1093.76002
[16]	Guo, B.-Y.; Yan, J.-P., Appl. Numer. Math., 59, 1386 (2009)
[17]	Doha, E. H.; Bhrawy, A. H.; Ezz-Eldien, S. S., Appl. Math. Model., 36, 4931 (2012)
[18]	Bhrawy, A. H.; Alofi, A. S., Commun. Nonlinear. Sci. Numer. Simulat., 17, 62 (2012) · Zbl 1244.65099
[19]	Doha, E. H.; Bhrawy, A. H.; Hafez, R. M., Commun. Nonlinear Sci. Numer. Simulat., 17, 3802 (2012)
[20]	Doha, E. H.; Bhrawy, A. H., Appl. Numer. Math., 58, 1224 (2008)
[21]	Zheng, Y.; Li, C.; Zhaoa, Z., Comput. Math. Appl., 59, 1718 (2010)
[22]	Su, L.; Wang, W.; Xu, Q., Appl. Math. Comput., 216, 3329 (2010)
[23]	Fix, G. J.; Roop, J. P., Comput. Math. Appl., 48, 1017 (2004)
[24]	Sousa, E., Comput. Math. Appl., 62, 938 (2011)
[25]	Tadjeran, C.; Meerschaert, M. M.; Scheffler, H.-P., J. Comput. Phys., 213, 205 (2006)
[26]	Abdel-Rehima, E. A.; Gorenflo, R., J. Comput. Appl. Math., 222, 274 (2008)
[27]	Zhang, H.; Liu, F.; Anh, V., Appl. Math. Comput., 217, 2534 (2010)
[28]	Guo, B.-Y.; Wang, L.-L., J. Appr. Theor., 128, 1-41 (2004)
[29]	Bhrawy, A. H.; Alofi, A. S.; Ezz-Eldien, S. S., Appl. Math. Lett., 24, 2146 (2011)
[30]	Sousa, E., Comput. Math. Appl., 64, 3141 (2012)
[31]	Ray, S. S.; Chaudhuri, K. S.; Bera, R. K., Appl. Math. Comput., 196, 294 (2008)