Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-24T06:08:43.008Z Has data issue: false hasContentIssue false
  • English
  • Français

Efficient Numerical Solution of the Multi-Term Time Fractional Diffusion-Wave Equation

Published online by Cambridge University Press:  06 March 2015

Jincheng Ren  and
Zhi-Zhong Sun
Jincheng Ren
Affiliation:
College of Mathematics and Information Science, Henan University of Economics and Law, Zhengzhou 450000, China
Zhi-Zhong Sun*
Affiliation:
Department of Mathematics, Southeast University, Nanjing 210096, China
*
*Corresponding author. Email addresses: [email protected] (J. Ren), [email protected] (Z. Z. Sun)
Get access

Abstract

Some efficient numerical schemes are proposed to solve one-dimensional and two-dimensional multi-term time fractional diffusion-wave equation, by combining the compact difference approach for the spatial discretisation and an L1 approximation for the multi-term time Caputo fractional derivatives. The unconditional stability and global convergence of these schemes are proved rigorously, and several applications testify to their efficiency and confirm the orders of convergence.

Keywords

65M0665M1265M15Multi-term time fractional diffusion-wave equationcompact difference schemediscrete energy methodconvergence
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 5 , Issue 1 , February 2015 , pp. 1 - 28
Copyright
Copyright © Global-Science Press 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Oldham, K.B. and Spanier, J., The Fractional Calculus, Academic Press, NewYork (1974).Google Scholar
[2]Samko, S.G., Kilbas, A.A. and Marichev, O.I., Fractional Integrals and Derivatives – Theory and Applications, Gordon and Breach Science Publishers (1993).Google Scholar
[3]Podlubny, I., Fractional Differential Equations, Academic Press, New York (1999).Google Scholar
[4]Mainardi, F., Fractional diffusive waves in viscoelastic solids, in Nonlinear Waves in Solids (Wegener, J.L. and Norwood, F.R., Eds.), pp. 93–97, ASME, Fairfield, New Jersey (1995).Google Scholar
[5]Metzler, R. and Klafter, J., Boundary value problems for fractional diffusion equations, Phys. A 278, 107–125 (2000).CrossRefGoogle Scholar
[6]Scher, H. and Montroll, E., Anomalous transit-time dispersion in amorphous solids, Phys. Rev. B 12, 2455 (1975).CrossRefGoogle Scholar
[7]Murillo, J.Q. and Yuste, S.B., An explicit difference method for solving fractional diffusion and diffusion-wave equations in the Caputo form, J. Comput. Nonlinear Dynamics 6, 021014 (2011).CrossRefGoogle Scholar
[8]Murillo, J.Q. and Yuste, S.B., On three explicit difference schemes for fractional diffusion and diffusion-wave equations, Phys. Scr. 136, 014025 (2009).Google Scholar
[9]Langlands, T. and Henry, B., The accuracy and stability of an implicit solution method for the fractional diffusion equation, J. Comp. Phys. 205, 719–736 (2005).CrossRefGoogle Scholar
[10]Chen, C., Liu, F., Turner, I. and Anh, V., A Fourier method for the fractional diffusion equation describing sub-diffusion, J. Comp. Phys. 227, 886–897 (2007).CrossRefGoogle Scholar
[11]Chen, C., Liu, F., Turner, I. and Anh, V., Numerical schemes and multivariate extrapolation of a two-dimensional anomalous sub-diffusion equation, Numer. Algor. 54, 1–21 (2010).CrossRefGoogle Scholar
[12]Sun, Z.Z. and Wu, X.N., A fully discrete difference scheme for a diffusion-wave system, Appl. Numer. Math. 56, 193–209 (2006).CrossRefGoogle Scholar
[13]Zhang, Y.N., Sun, Z.Z. and Zhao, X., Compact alternating direction implicit schemes for the two-dimensional fractional diffusion-wave equation, SIAM J. Numer. Anal. 50, 1535–1555 (2012).CrossRefGoogle Scholar
[14]Diethelm, K. and Luchko, Y., Numerical solution of linear multi-term differential equations of fractional order, J. Comp. Anal. Appl. 6, 243–263 (2004).Google Scholar
[15]Lubich, C., Discretized fractional calculus, SIAM J. Math. Anal. 17, 704–719 (1986).CrossRefGoogle Scholar
[16]Edwards, J.T., Neville, J.F. and Simpson, A.C., The numerical solution of linear multi-term fractional differential equations: systems of equations, J. Comp. Anal. Appl. 148), 401–418 (2002).Google Scholar
[17]Katsikadelis, J.T., Numerical solution of multi-term fractional differential equations, Z. Angew. Math. Mech. 89, 593–608 (2009).CrossRefGoogle Scholar
[18]Jafari, H., Golbabai, A., Seifi, S. and Sayevand, K., Homotopy analysis method for solving multi-term linear and nonlinear diffusion-wave equations of fractional order, Comput. Math. Appl. 59, 1337–1344 (2010).CrossRefGoogle Scholar
[19]Daftardar-Gejji, V. and Bhalekar, S., Boundary value problems for multi-term fractional differential equations, J. Math. Anal. Appl. 345, 754–765 (2008).CrossRefGoogle Scholar
[20]Zhang, Y.N., Sun, Z.Z. and Wu, H.W., Error estimates of Crank-Nicolson-type difference scheme for the subdiffusion equation, SIAM J. Numer. Anal. 49, 2302–2322 (2011).CrossRefGoogle Scholar
[21]Zhou, Y.L., Applications of Discrete Functional Analysis to Finite Difference Method, International Academic Publishers, Beijing (1990).Google Scholar
[22]Sun, Z.Z., Numerical Methods of Partial Differential Equations, 2nd Edition (in Chinese), Science Press, Beijing, (2012).Google Scholar
[23]Gao, G.H. and Sun, Z.Z., A compact finite difference scheme for the fractional sub-diffusion equations, J. Comp. Phys. 230, 586–595 (2011).CrossRefGoogle Scholar
[24]Sun, Z.Z., Compact difference schemes for the heat equation with Neumann boundary conditions, Numer. Meth. Partial Differential Equations 25, 1320–1341 (2009).CrossRefGoogle Scholar
[25]Sun, Z.Z., The Method of Order Reduction and Its Application to the Numerical Solutions of Partial Differential Equations, Science Press, Beijing (2009).Google Scholar
[26]Ren, J.C. and Sun, Z.Z., Numerical algorithm with high spatial accuracy for the fractional diffusion-wave equation with Neumann boundary conditions, J. Sci. Comp. 56, 381–408 (2013).CrossRefGoogle Scholar
[27]Ren, J.C. and Sun, Z.Z., Efficient and stable numerical methods for multi-term time-fractional sub-diffusion equations, East Asian J. Appl. Math. 4, 242–266 (2014).CrossRefGoogle Scholar
[28]Conte, S.D. and Boor, C. de, Elementary Numerical Analysis: An Algorithmic Approach, 3rd Edition, McGraw-Hill, New York (1980).Google Scholar