# Implicit finite difference approximation for time fractional heat conduction under boundary condition of second kind

## DOI:

https://doi.org/10.14419/ijamr.v4i1.3783## Published:

2015-01-29## Keywords:

Boundary condition of second kind, Caputo fractional derivative, Implicit finite difference scheme, Time fractional heat conduction, Unconditionally stable.## Abstract

The time fractional heat conduction in an infinite plate of finite thickness, when both faces are subjected to boundary conditions of second kind, has been studied. The time fractional heat conduction equation is used, when attempting to describe transport process with long memory, where the rate of heat conduction is inconsistent with the classical Brownian motion. The stability and convergence of this numerical scheme has been discussed and observed that the solution is unconditionally stable. The whole analysis is presented in dimensionless form. A numerical example of particular interest has been studied and discussed in details.

## References

[1] D. Benson, S. Wheatcraft, M. Meerschaert, Application of fractional advection-dispersion equation, Water Resour. Res., 36 (6) (2000) 1403-1412.

[2] C. Chen, F. Liu, K. Burrage, Finite difference methods and a Fourier analysis for the fractional reaction-sub diffusion equation, Applied Mathematics and Computation, 198 (2) (2008) 754-769.

[3] C. Chen, F. Liu, I. Turner, V. Anh, A Fourier method for the fractional diffusion equation describing sub-diffusion, J. Comput. Phys., 227 (2) (2007) 886-897.

[4] H. Ding, Y. Zhang, Notes on implicit finite difference approximation for time fractional diffusion equations, Int. J. Compu. Math. Appl., 61 (2011) 2924-2928.

[5] P. Grassberger, W. Nadler, L. Yang, Heat conduction and entropy production in a one-dimensional hard-particle gas, Physical Rev. Lett., 89 (18) (2002) 1806011-1806014.

[6] F. Huang, F. Liu, The time fractional diffusion and advection dispersion equation, ANZIAM J. 46 (2005) 317-330.

[7] X. Jiang, M. Xu, The time fractional heat conduction equation in the general orthogonal curvilinear coordinate and the cylindrical coordinate systems, Physica A, 389 (2010) 3368-3374.

[8] F. Khanna, D. Matrasulov, Non-linear Dynamics and Fundamental Interactions, Nato Public Diplomacy Division, 2004.

[9] T. Langlands, B. Henry, The accuracy and stability of an implicit solution method for the fractional diffusion equation, J. Comput. Phys. 205 (2005) 719-736.

[10] S. Lepri, R. Livi, A. Politi, On the anomalous thermal conductivity of one-dimensional lattices, Europhys. Lett., 43 (3) (1998) 271-276.

[11] F. Mainardi, M. Raberto, R. Goreno, E. Scalas, Fractional calculus and continuous-time finance II: the waiting-time distribution, Physica A, 287 (2000) 468-481.

[12] M. Meerschaert, D. Benson, B. Baeumer, Operator levy motion and multi-scaling anomalous diffusion, Physical Rev. E, 63 (2001) 11121-11126.

[13] M. Meerschaert, D. Benson, H. Scheer, B.Baeumer, Stochastic solution of space-time fractional diffusion equations, Physical Rev. E 65 (2002) 11031-11034.

[14] D. Murio, Implicit finite difference approximation for time fractional diffusion equations, Int. J. of Compu. Math. Appl., 56 (2008) 1138-1145.

[15] M. Naber, Distributed order fractional sub-diffusion, Fractals, 12 (23) (2004) 23-32.

[16] F. Norwood, Transient thermal waves in the general theory of heat conduction with finite wave speeds, J. Appl. Mech., 39 (3) (1972) 673-676.

[17] M. Ozisik, Heat Conduction, John Wiley & Sons, Inc, New York, 1993.

[18] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.

[19] Y. Povstenko, Fractional heat conduction equation and associated thermal stresses in an infinite solid with spherical cavity, Quarterly Journal of Mechanics and Applied Mathematics, 61 (4) (2008) 523-547.

[20]Y. Povstenko, Signaling problem for time fractional diffusion-wave equation in a half-space in the case of angular symmetry, Nonlinear Dyn., 59 (2010) 593-605.

[21]M. Raberto, E. Scalas, F. Mainardi, Waiting-times and returns in high frequency financial data: an empirical study, Physica A, 314 (2002) 749-755.

[22] Z. Rieder, J. Lebowitz, E. Lieb, Properties of harmonic crystal in stationary nonequilibrium state, J. Mathematical Phys., 8 (1967) 1073-1078.

[23] J. Singh, P. Gupta, K. Rai, Solution of fractional bioheat equations by finite difference method and HPM, Mathematical and Computer Modelling, 54 (2011) 2316-2325.

[24] T. Solomon, E.Weeks, H. Swinney, Observation of anomalous diffusion and levy flights in a 2-dimensional rotating flow, Physical Rev. Lett., 71 (24) (1993) 3975-3978.

[25] S. Yuste, Weighted average finite difference methods for fractional diffusion equations, J. Comput. Phys., 216 (2006) 264-274.

[26] S. Yuste, L. Acedo, An explicit finite difference method and a new von Neumann-type stability analysis for fractional diffusion equations, SIAM J. Numer. Anal., 42 (5) (2005) 1862 1874.

[27] S. Yuste, K. Lindenberg, Sub-diffusion-limited A+A reactions, Physical Rev. Lett., 87 (2001) 118301-118304.

[28] Y. Zhang, A finite difference method for fractional partial differential equation, Applied Mathematics and Computation, 215 (2009) 524-529.

[29] X. Zhao, Z. Sun, A box-type scheme for the fractional sub-diffusion equation with Neumann boundary conditions, J. Comput. Phys., 230 (2011) 6061-6074.

[30] P. Zhuang, F. Liu, Implicit difference approximation for the time fractional diffusion equation, Journal of Applied Mathematics and Computing, 22 (3) (2006) 87-99.

[31] P. Zhuanga, F. Liu, V. Anh, I. Turner, New solution and analytical techniques of the implicit numerical method for the anomalous sub-diffusion equation, SIAM J. Numer. Anal., 46 (2) (2008) 1079-1095.

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