Commun. Comput. Phys., 13 (2013), pp. 502-525.

Numerical Solution for a Non-Fickian Diffusion in a Periodic Potential

Aderito Araujo 1, Amal K. Das 2, Cidalia Neves 3, Ercilia Sousa 1*

1 CMUC, Department of Mathematics, University of Coimbra, 3001-454 Coimbra, Portugal.
2 Department of Physics, Dalhousie University, Halifax, Nova Scotia B3H 3J5, Canada.
3 CMUC, Department of Mathematics, University of Coimbra, 3001-454 Coimbra, Portugal; and ISCAC, Polytechnic Institute of Coimbra, 3040-316 Coimbra, Portugal.

Received 28 July 2011; Accepted (in revised version) 1 March 2012
Available online 17 July 2012


Numerical solutions of a non-Fickian diffusion equation belonging to a hyperbolic type are presented in one space dimension. The Brownian particle modelled by this diffusion equation is subjected to a symmetric periodic potential whose spatial shape can be varied by a single parameter. We consider a numerical method which consists of applying Laplace transform in time; we then obtain an elliptic diffusion equation which is discretized using a finite difference method. We analyze some aspects of the convergence of the method. Numerical results for particle density, flux and mean-square-displacement (covering both inertial and diffusive regimes) are presented.

AMS subject classifications: 35L15, 60K40, 60J65, 65R10, 65M06, 65M12
PACS: 02.60.Lj, 05.40.Jc
Key words: Numerical methods, Laplace transform, telegraph equation, periodic potential, non-Fickian diffusion.

*Corresponding author.
Email: (A. Araujo), (A. K. Das), (C. Neves), (E. Sousa)

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