NUMERICAL SOLUTIONS FOR SPACE FRACTIONAL DISPERSION EQUATIONS WITH NONLINEAR SOURCE TERMS

- Journal title : Bulletin of the Korean Mathematical Society
- Volume 47, Issue 6, 2010, pp.1225-1234
- Publisher : The Korean Mathematical Society
- DOI : 10.4134/BKMS.2010.47.6.1225

Title & Authors

NUMERICAL SOLUTIONS FOR SPACE FRACTIONAL DISPERSION EQUATIONS WITH NONLINEAR SOURCE TERMS

Choi, Hong-Won; Chung, Sang-Kwon; Lee, Yoon-Ju;

Choi, Hong-Won; Chung, Sang-Kwon; Lee, Yoon-Ju;

Abstract

Numerical solutions for the fractional differential dispersion equations with nonlinear forcing terms are considered. The backward Euler finite difference scheme is applied in order to obtain numerical solutions for the equation. Existence and stability of the approximate solutions are carried out by using the right shifted Grunwald formula for the fractional derivative term in the spatial direction. Error estimate of order is obtained in the discrete norm. The method is applied to a linear fractional dispersion equations in order to see the theoretical order of convergence. Numerical results for a nonlinear problem show that the numerical solution approach the solution of classical diffusion equation as fractional order approaches 2.

Keywords

fractional differential equation;Riemann-Liouville fractional derivative;Caputo fractional derivative;finite difference scheme;stability;convergence;error estimate;

Language

English

Cited by

1.

2.

3.

4.

5.

6.

References

1.

B. Baeumer, M. Kovacs, and M. M. Meerschaert, Fractional reproduction-dispersal equations and heavy tail dispersal kernels, Bull. Math. Biol. 69 (2007), no. 7, 2281-2297.

2.

B. Baeumer, M. Kovacs, and M. M. Meerschaert, Numerical solutions for fractional reaction-diffusion equations, Comput. Math.
Appl. 55 (2008), no. 10, 2212-2226.

3.

V. J. Ervin, N. Heuer, and J. P. Roop, Numerical approximation of a time dependent, nonlinear, space-fractional diffusion equation, SIAM J. Numer. Anal. 45 (2007), no. 2, 572-591.

4.

V. E. Lynch, B. A. Carreras, D. del-Castillo-Negrete, K. M. Ferreira-Mejias, and H. R. Hicks, Numerical methods for the solution of partial differential equations of fractional order, J. Comput. Phys. 192 (2003), no. 2, 406-421.

5.

M. M. Meerschaert and C. Tadjeran, Finite difference approximations for fractional advection-dispersion flow equations, J. Comput. Appl. Math. 172 (2004), no. 1, 65-77.

6.

M. M. Meerschaert and C. Tadjeran, Finite difference approximations for two-sided space-fractional partial differential equations, Appl. Numer. Math. 56 (2006), no. 1, 80-90.

7.

M. M. Meerschaert, H.-P. Scheffler, and C. Tadjeran, Finite difference methods for two-dimensional fractional dispersion equation, J. Comput. Phys. 211 (2006), no. 1, 249-261.

8.

K. B. Oldham and J. Spanier, The Fractional Calculus, Dover Publications, New York,
2002.

9.

J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several
Variables, Academic Press, New York-London, 1970.

10.

J. P. Roop, Computational aspects of FEM approximation of fractional advection dispersion equations on bounded domains in $R^2$ , J. Comput. Appl. Math. 193 (2006), no. 1, 243-268.

11.

C. Tadjeran, M. M. Meerschaert, and H.-P. Scheffler, A second-order accurate numerical
approximation for the fractional diffusion equation, J. Comput. Phys. 213 (2006), no.
1, 205-213.

12.

H. Ye, J. Gao, and Y. Ding, A generalized Gronwall inequality and its application to a
fractional differential equation, J. Math. Anal. Appl. 328 (2007), no. 2, 1075-1081.

13.

R. S. Varga, Matrix Iterative Analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962.