Keywords:-

Keywords: time fractional, soil moisture diffusion equation, finite difference scheme, stability analysis, convergence analysis, fractional derivatives, software Mathematica.

Article Content:-

Abstract

In this paper, we give a detailed analysis for the explicit finite difference approximation for time fractional soil moisture diffusion equation (TFSMDE). Furthermore, the stability and convergence of the scheme in a bounded domain are discussed. As an application of the scheme we solve some test problems and their solutions are represented graphically by a powerful software Mathematica.

References:-

References

O. P. Agrawal, Solution for a Fractional Diffusion-Wave Equation Defined in

a Bounded Domain, J. Nonlinear Dynamics 29, (2002).

D. Baleanu,J.A. Machado and A.C.J. Luo, Fraction Dynamics and Control,

Springer, New York Dordrecht Heidelberg, London, (2012).

A. P. Bhadane and K. C. Takale, Basic Developments of Fractional Calculus

and its Applications, Bulletin of Marathwada Mathematical Society, Vol.

,(2011).

Daniel Hillel, Introduction to Soil Physics, Academic Press (1982).

M. K. Datar, S. R. Kulkarni and K. C. Takale, Finite Difference Approximation

for Black-Scholes Partial Differential Equation and Its Application, Int

Jr. of Mathematical Sciences and Applications, Vol 3,(2013).

D. B. Dhaigude and K. C. Takale, Two Level Explicit Finite Difference

Scheme for One-Dimensional Penne’s Bioheat Equation, Recent Advances in

Mathematical Sciences and Applications, p. 230-238,(2009)

Don Kirkham and W.L. Powers, Advanced Soil Physics, Wiley-Interscience

(1971).

G. Gao and Z. Sun, A compact finite difference scheme for the fractional

sub-diffusion equations, J. Comput. Phys., 230:586-595,(2011).

R. Goren

o and F. Mainardi, Approximation of Levy-Feller Diffusion

by Random Walk, Journal for Analysis and its Applications (ZAA) 18:231-

,(1999).

R. Goren

o, F. Mainardi, D. Moretti and P. Paradisi, Time Fractional

Diffusion: A Discrete Random Walk Approach, Nonlinear Dynamics 29, (2002)

R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore

(2000).

A. A. Kilbas, H. M. Srivastava and J. J. Trijillo, Theory and Applications

of Fractional Differential Equations, North-Holland Mathematical Studies, Vol

,(2006).F. Liu, V. Anh, I. Turner and P. Zhuang, Numerical Simulation for solute

transport in fractal porous media, ANZIAM J. 45(E),(2004).

Y. Lin, C. Xu, Finite difference/spectral approximations for the time-fractional

diffusion equation, J. Comput. Phys. 225 , pp.1533-1552:(2007)

M.M. Meerschaert, C. Tadjeran, Finite difference approximations for twosided

space-fractional partial differential equations, Appl. Numer. Math. 56:80-

,(2006)

M.M. Meerschaert, C. Tadjeran, Finite difference approximations for

fractional advection-dispersion flow equations, J. Comput. Appl. Math. 172:65-

,(2004).

R. Metzler and J. Krafter, The Random Walks Guide to Anomalous Diffusion:

A Fractional Dynamic Approach, Physical Reports 339, (2000).

K. S. Miller, B. Ross, An introduction to the Fractional Calculus and Fractional

Differential Equations, John Wiley and Sons, New York,(1993).

Momani S, Odibat Z, Numerical comparison of methods for solving linear

differential equations of fractional order, Chaos Soliton Fract 31:12481255,(2007).

Momani S, Odibat Z, Numerical approach to differential equations of fractional

order, J Comp Appl Math 207(1):96110, (2007).

Pater A.C., Raats and Martinus TH. Ven Genuchten, Milestones in Soil

Physics, J. Soil Science 171,1(2006).

I. Podlubny, Fractional Differential equations, Academic Press, San Diago

(1999).

M. Rehman, R. A. Khan, A numerical method for solving boundary value

problems for fractional differential equations, Appl. Math. Model., 36(3), 894-

, (2012).

B.Ross (Ed), Fractional Calculus and Its Applications, Lecture Notes in Mathematics,

Vol.457, Springer-Verlag, New York ( 1975).

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals

and Derivatives: Theory and Applications, Gordan and Breach, Newark, N. J.,

(1993).

S. Shen, F.Liu, Error Analysis of an explicit Finite Difference Approximation

for the Space Fractional Diffusion equation with insulated ends, ANZIAM J.46

(E), pp. C871 - C887: (2005).

G.D. Smith, Numerical Solution of Partial Differential Equations, (2ndEdn.),

Clarendon Press, Oxford, (1978).K. C. Takale and A. S. Shinde, Study of Black-Scholes Model and its Applications,

Elsevier Ltd., Journal ”Procedia Engineering” , Vol. 38, (2012).

W. Wyss and W. R. Schneider, Fractional Diffusion and wave equations, J.

Math. Phys. 30,(1989).

W. Wyss, The Fractional Diffusion Equation, J. Math. Phys. 27,(1986).

Downloads

Citation Tools

How to Cite
Sontakke, B. R., & Sangvikar, V. eena V. (2016). ANALYSIS OF EXPLICIT DIFFERENCE APPROXIMATE SOLUTION FOR TIME FRACTIONAL SOIL MOISTURE DIFFUSION EQUATION. International Journal Of Mathematics And Computer Research, 4(05), 1387-1400. Retrieved from http://ijmcr.in/index.php/ijmcr/article/view/46