The new Implicit Fi nite Difference Method for the Solution of Time Fractional Advection-Dispersion Equation
محورهای موضوعی : Applied Mathematics
Hamid Reza Khodabandehloo
1
(Department of Mathematics, Payame Noor University (PNU),45771-13878,
Qeydar, Zanjan, Iran)
Elyas Shivanian
2
(Department of Mathematics, Imam Khomeini International University,
Qazvin, Iran)
Sh. Mostafaee
3
(Department of Mathematics, Imam Khomeini International University,
Qazvin, Iran)
کلید واژه:
چکیده مقاله :
In this paper, a numerical solution of time fractional advection-dispersion equations are presented.The new implicit nite difference methods for solving these equations are studied. We examinepractical numerical methods to solve a class of initial-boundary value fractional partial differentialequations with variable coefficients on a nite domain. Stability, consistency, and (therefore) convergenceof the method are examined and the local truncation error is O(Δt + h). This study concernsboth theoretical and numerical aspects, where we deal with the construction and convergence analysisof the discretization schemes. The results are justied by some numerical implementations. Anumerical example with known exact solution is also presented, and the behavior of the error isexamined to verify the order of convergence.
[1]Goreno R., Mainardi F., Scalas E., Raberto M., Fractional calculus and
continuous-time nance. III, The di usion limit. Mathematical nance
(Konstanz, 2000), Trends in Math., Birkhuser, Basel, 2001, pp. 171-180.
[2]Lubich C., Discretized fractional calculus, SIAM J. Math. Anal.17 (1986)
704719.13.
[3]Meerschaert M. M. , Tadjeran C ., Finite di erence approximations for
fractional advection - di usion ow equations, J.comput. Appl. Numer.
Math.172 (2004) 6577.
[4]Podlubny I., Fractional Di erential Equations, Academic Press, New York,
1999.
[5]Samko S., Kilbas A., Marichev O., Fractional Integrals and Derivatives:
Theory and Applications, Gordon and Breach, London, 1993.
[6]Oldham K.B., Spanier J., The Fractional Calculus, Academic Press, New
York, 1974.
[7]Tadjeran C., Meerschaert M. M., H.P. Sche er, A second-order accurate
numerical approximation for the fractional di usion equation, J. Comput.
Phys. 213 (2006) 205-213.
[8]Miller K., Ross B., An Introduction to the Fractional Calculus and
Fractional Di erential, Wiley, New York, 1993.
[9]zhang Y., A Finite di erence method for fractional partial di erential
equation, Appl. Math. comput. 215 (2009) 524-529.
