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Engineering with Computers

, Volume 35, Issue 1, pp 229–241 | Cite as

Semi-analytical solution for time-fractional diffusion equation based on finite difference method of lines (MOL)

  • Saeed Kazem
  • Mehdi DehghanEmail author
Original Article

Abstract

In this article, we apply the method of lines (MOL) for solving the time-fractional diffusion equations (TFDEs). The use of MOL yields a system of fractional differential equations with the initial value. The solution of this system could be obtained in the form of Mittag–Leffler matrix function. A direct method which computes the Mittag–Leffler matrix by applying its eigenvalues and eigenvectors analytically has been discussed. The direct approach has been applied on one-, two-, and three-dimensional TFDEs with Dirichlet, Neumann, and periodic boundary conditions as well.

Keywords

Method of lines Time-fractional diffusion equations Mittag–Leffler function Matrix exponential function Tridiagonal matrix Dirichlet Neumann and periodic boundary conditions 

AMS subject classification:

65M20 65M06 

Notes

Acknowledgements

The authors are grateful to the reviewers for their comments and suggestions which have improved the paper.

References

  1. 1.
    Bellman R (1997) Introduction to matrix analysis. Society for Industrial and Applied Mathematics. Philadelphia, PAzbMATHGoogle Scholar
  2. 2.
    Bini DA, Gemignani L, Tisseur F (2005) The Ehrlich-Aberth method for the nonsymmetric tridiagonal eigenvalue problem. SIAM J Matrix Anal Appl 27:153–175MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Diethelm K (2010) The analysis of fractional differential equations, an application oriented exposition using differential operators of Caputo type. Springer, BerlinzbMATHGoogle Scholar
  4. 4.
    Dehghan M, Manafian J, Saadatmandi A (2010) Solving nonlinear fractional partial differential equations using the homotopy analysis method. Numer Methods Part Differ Equ 26(2):448–479MathSciNetzbMATHGoogle Scholar
  5. 5.
    Dehghan M, Shakeri F (2009) Method of lines solutions of the parabolic inverse problem with an overspecification at a point. Numer Algor 50:417–437MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Dehghan M, Mohammadi V (2015) The method of variably scaled radial kernels for solving two-dimensional magnetohydrodynamic (MHD) equations using two discretizations: The Crank-Nicolson scheme and the method of lines (MOL). Comput Math Appl 70(10):2292–2315MathSciNetCrossRefGoogle Scholar
  7. 7.
    Esmaeili S (2017) Solving 2D time-fractional diffusion equations by a pseudospectral method and Mittag-Leffler function evaluation. Math Methods Appl Sci 40:1838–1850MathSciNetzbMATHGoogle Scholar
  8. 8.
    Euler L (1730) Memoire dans le tome V des Comment. Saint Petersberg Annees, 55Google Scholar
  9. 9.
    Garrappa R (2015) numerical evaluation of two and three parameter Mittag-Leffler functions. SIAM J Numer Anal 53:1350–1369MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Garrappa R, Popolizio M (2013) Evaluation of generalized Mittag-Leffler functions on the real line. Adv Comput Math 39:205–225MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Heath MT (2002) Scientific computing: an introductory survey, 2nd edn. McGraw-Hill, New YorkzbMATHGoogle Scholar
  12. 12.
    Higham NJ (2008) Functions of matrices theory and computation, vol 104. Society for Industrial and Applied Mathematics, Philadelphia, PACrossRefzbMATHGoogle Scholar
  13. 13.
    Higham NJ (2005) The scaling and squaring method for the matrix exponential revisited. SIAM J Matrix Anal Appl 26:1179–1193MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Hundsdorfer W, Verwer JG (2003) Numerical solution of time-dependent advection-diffusion-reaction equations, vol 33. Springer, BerlinzbMATHGoogle Scholar
  15. 15.
    Kazem S (2013) Exact solution of some linear fractional differential equations by Laplace transform. Int J Nonlinear Sci 16:3–11MathSciNetzbMATHGoogle Scholar
  16. 16.
    Khan Y, Wu Q, Faraz N, Yildirim A, Madani M (2012) A new fractional analytical approach via a modified Riemann-Liouville derivative. Appl Math Lett 25:1340–1346MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Khan Y, Faraz N, Kumar S, Yildirim A (2012) A Coupling method of homotopy perturbation and Laplace transformation for fractional models. UPB Sci Bull Ser A 74:57–68MathSciNetzbMATHGoogle Scholar
  18. 18.
    Khan Y, Ali Beik SP, Sayevand K, Shayganmanesh A (2015) A numerical scheme for solving differential equations with space and time-fractional coordinate derivatives. Quaest Math 38:41–55MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Khan Y, Sayevand K, Fardi M, Ghasemi M (2014) A novel computing multi-parametric homotopy approach for system of linear and nonlinear Fredholm integral equations. Appl Math Comput 249:229–236MathSciNetzbMATHGoogle Scholar
  20. 20.
    Lombardi G, Rebaudo R (1988) Eigenvalues and eigenvectors of a special class of band matrices. Rend Ist Mat Univ Trieste 20:113–128MathSciNetzbMATHGoogle Scholar
  21. 21.
    Mitchell AR, Griffiths DF (1980) The finite diffrerence method in partial differential equations. John Wiley, New YorkGoogle Scholar
  22. 22.
    Moret I, Novation P (2011) the convergence of Krylov subspace methods for matrix Mittag-Leffler functions. SIAM J Numer Anal 49:2144–2164MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Norsett SP, Wolfbrandt A (1977) Attainable order of rational approximations to the exponential function with only real poles. BIT 17:200–208MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Podlubny I (1999) Fractional differential equations. Academic Press, San DiegozbMATHGoogle Scholar
  25. 25.
    Podlubny I, Kacenak M (2005) Mittag-Leffler function; calculates the Mittag-Leffler function with desired accuracy , MATLAB Central File Exchange, File ID 8735, mlf. mGoogle Scholar
  26. 26.
    Reusch MF, Ratzan L, Pomphrey N, Park W (1988) Diagonal Padé approximations for initial value problems. SIAM J Sci Stat Cornput 9:829–838CrossRefzbMATHGoogle Scholar
  27. 27.
    Saadatmandi A, Dehghan M (2010) A new operational matrix for solving fractional-order differential equations. Comput Math Appl 59(3):1326–1336MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Serbin SM (1992) A scheme for parallelizing certain algorithms for the linear inhomogeneous heat equation. SIAM J Sci Stat Cornput 13:449–458MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Shakeri F, Dehghan M (2008) The method of lines for solution of the one-dimensional wave equation subject to an integral conservation condition. Comput Math Appl 56:2175–2188MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Sun ZZ, Wu X (2006) A fully discrete difference scheme for a diffusion-wave system. Appl Numer Math 56:193–209MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Zakian V (1971) Comment on rational approximations to the matrix exponential. Electron Lett 7:261–262MathSciNetCrossRefGoogle Scholar
  32. 32.

Copyright information

© Springer-Verlag London Ltd., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Applied Mathematics, Faculty of Mathematics and Computer ScienceAmirkabir University of TechnologyTehranIran

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