Advertisement

Engineering with Computers

, Volume 35, Issue 1, pp 75–86 | Cite as

A haar wavelet approximation for two-dimensional time fractional reaction–subdiffusion equation

  • Ömer Oruç
  • Alaattin Esen
  • Fatih Bulut
Original Article

Abstract

In this study, we established a wavelet method, based on Haar wavelets and finite difference scheme for two-dimensional time fractional reaction–subdiffusion equation. First by a finite difference approach, time fractional derivative which is defined in Riemann–Liouville sense is discretized. After time discretization, spatial variables are expanded to truncated Haar wavelet series, by doing so a fully discrete scheme obtained whose solution gives wavelet coefficients in wavelet series. Using these wavelet coefficients approximate solution constructed consecutively. Feasibility and accuracy of the proposed method is shown on three test problems by measuring error in \(L_{\infty }\) norm. Further performance of the method is compared with other methods available in literature such as meshless-based methods and compact alternating direction implicit methods.

Keywords

Two-dimensional Haar wavelets Two-dimensional reaction–subdiffusion Fractional two-dimensional problem Numerical solution 

Mathematics Subject Classification

65T60 65M70 35R11 

References

  1. 1.
    Kilbas AA, Srivastava HM, Trujillo JJ (2006) Theory and applications of fractional differential equations. Elsevier, AmsterdamzbMATHGoogle Scholar
  2. 2.
    Ray SS (2007) Exact solutions for time-fractional diffusion-wave equations by decomposition method. Phys Scr 75:53–61MathSciNetzbMATHGoogle Scholar
  3. 3.
    Saadatmandi A, Dehghan M, Azizi MR (2012) The Sinc–Legendre collocation method for a class of fractional convection diffusion equation with variable coefficients. Commun Nonlinear Sci Numer Simul 17(11):4125–4136MathSciNetzbMATHGoogle Scholar
  4. 4.
    Saadatmandi A, Dehghan M (2010) A new operational matrix for solving fractional-order differential equations. Comput Math Appl 59(3):1326–1336MathSciNetzbMATHGoogle Scholar
  5. 5.
    Dehghan M, Manafian J, Saadatmandi A (2010) Solving nonlinear fractional partial differential equations using the homotopy analysis method. Numer Methods Partial Differ Equ 26(2):448–479MathSciNetzbMATHGoogle Scholar
  6. 6.
    Yousefi SA, Lotfi A, Dehghan M (2011) The use of Legendre multiwavelet collocation method for solving the fractional optimal control problems. J Vib Control 17(13):2059–2065MathSciNetzbMATHGoogle Scholar
  7. 7.
    Agrawal OP (2002) Solution for a fractional diffusion-wave equation defined in a bounded domain. Nonlinear Dyn 29:145–155MathSciNetzbMATHGoogle Scholar
  8. 8.
    Esen A, Ucar Y, Yagmurlu N, Tasbozan O (2013) A Galerkin finite element method to solve fractional diffusion and fractional diffusion-wave equations. Math Model Anal 18:260–273MathSciNetzbMATHGoogle Scholar
  9. 9.
    Mohebbi A, Mostafa A, Dehghan M (2013) The use of a meshless technique based on collocation and radial basis functions for solving the time fractional nonlinear Schrödinger equation arising in quantum mechanics. Eng Anal Bound Elem 37:475–485MathSciNetzbMATHGoogle Scholar
  10. 10.
    Hosseini VR, Chen W, Avazzade Z (2014) Numerical solution of fractional telegraph equation by using radial basis functions. Eng Anal Bound Elem 38:31–39MathSciNetGoogle Scholar
  11. 11.
    Wei L, Dai H, Zhang D, Si Z (2014) Fully discrete local discontinuous Galerkin method for solving the fractional telegraph equation. Calcolo 51:175–192MathSciNetzbMATHGoogle Scholar
  12. 12.
    Meerschaert MM, Scheffler HP, Tadjeran C (2006) Finite difference methods for two-dimensional fractional dispersion equation. J Comput Phys 211(1):249–261MathSciNetzbMATHGoogle Scholar
  13. 13.
    Tadjeran C, Meerschaert MM (2007) A second-order accurate numerical method for the two-dimensional fractional diffusion equation. J Comput Phys 220(2):813–823MathSciNetzbMATHGoogle Scholar
  14. 14.
    Zhuang P, Liu F (2007) Finite difference approximation for two-dimensional time fractional diffusion equation. J Algorithms Comput Technol 1(1):1–15Google Scholar
  15. 15.
    Chen S, Liu F (2008) ADI-Euler and extrapolation methods for the two-dimensional fractional advection dispersion equation. J Appl Math Comput 26(1–2):295–311MathSciNetzbMATHGoogle Scholar
  16. 16.
    Chen CM, Liu F, Turner I, Anh V (2010) Numerical schemes and multivariate extrapolation of a two dimensional anomalous sub-diffusion equation. Numer Algorithms 54(1):1–21MathSciNetzbMATHGoogle Scholar
  17. 17.
    Chen CM, Liu F, Anh V, Turner I (2011) Numerical methods for solving a two-dimensional variable order anomalous subdiffusion equation. Math Comput 81(277):345–366MathSciNetzbMATHGoogle Scholar
  18. 18.
    Zhang YN, Sun ZZ, Zhao X (2012) Compact alternating direction implicit scheme for the two-dimensional fractional diffusion-wave equation. SIAM J Numer Anal 50(3):1535–1555MathSciNetzbMATHGoogle Scholar
  19. 19.
    Cui MR (2013) Convergence analysis of high-order compact alternating direction implicit schemes for the two-dimensional time fractional diffusion equation. Numer Algorithms 62(3):383–409MathSciNetzbMATHGoogle Scholar
  20. 20.
    Zeng F, Liu F, Li CP, Burrage K, Turner I, Anh V (2014) A Crank–Nicolson ADI spectral method for a two-dimensional Riesz space fractional nonlinear reaction–diffusion equation. SIAM J Numer Anal 52:2599–2622MathSciNetzbMATHGoogle Scholar
  21. 21.
    Abbaszadeh M, Dehghan M (2015) A meshless numerical procedure for solving fractional reaction subdiffusion model via a new combination of alternating direction implicit (ADI) approach and interpolating element free Galerkin (EFG) method. Comput Math Appl 70:2493–2512MathSciNetGoogle Scholar
  22. 22.
    Dehghan M, Abbaszadeh M, Mohebbi A (2015) Error estimate for the numerical solution of fractional reaction–subdiffusion process based on a meshless method. J Comput Appl Math 280:14–36MathSciNetzbMATHGoogle Scholar
  23. 23.
    Yu B, Jiang X, Xu H (2015) A novel compact numerical method for solving the two-dimensional non-linear fractional reaction–subdiffusion equation. Numer Algorithms 68:923–950MathSciNetzbMATHGoogle Scholar
  24. 24.
    Lepik U (2007) Application of the Haar wavelet transform to solving integral and differential equations. Proc Estonian Acad Sci Phys Math 56:28–46MathSciNetzbMATHGoogle Scholar
  25. 25.
    Chen C, Hsiao CH (1997) Haar wavelet method for solving lumped and distributed parameter systems. IEE Proc Control Theory Appl 144:87–94zbMATHGoogle Scholar
  26. 26.
    Chen C, Hsiao CH (1997) Wavelet approach to optimising dynamic systems. IEE Proc Control Theory Appl 146:213–219Google Scholar
  27. 27.
    Hsiao CH, Wang WJ (2000) State analysis of time-varying singular bilinear systems via Haar wavelets. Math Comput Simul 52:11–20MathSciNetGoogle Scholar
  28. 28.
    Hsiao CH, Wang WJ (1999) State analysis of time-varying singular nonlinear systems via Haar wavelets. Math Comput Simul 51:91–100MathSciNetGoogle Scholar
  29. 29.
    Hsiao CH, Wang WJ (2001) Haar wavelet approach to nonlinear stiff systems. Math Comput Simul 57:347–353MathSciNetzbMATHGoogle Scholar
  30. 30.
    Hsiao CH (2004) Haar wavelet direct method for solving variational problems. Math Comput Simul 64:569–585MathSciNetzbMATHGoogle Scholar
  31. 31.
    Lepik U (2005) Numerical solution of differential equations using Haar wavelets. Math Comput Simul 68:127–143MathSciNetzbMATHGoogle Scholar
  32. 32.
    Lepik U (2007) Numerical solution of evolution equations by the Haar wavelet method. Appl Math Comput 185:695–704MathSciNetzbMATHGoogle Scholar
  33. 33.
    Lepik U (2011) Solving PDEs with the aid of two-dimensional Haar wavelets. Comput Math Appl 61:1873–1879MathSciNetzbMATHGoogle Scholar
  34. 34.
    Jiwari R (2012) A Haar wavelet quasilinearization approach for numerical simulation of Burgers’ equation. Comput Phys Commun 183:2413–2423MathSciNetzbMATHGoogle Scholar
  35. 35.
    Jiwari R (2015) A hybrid numerical scheme for the numerical solution of the Burgers’ equation. Comput Phys Commun 188:59–67MathSciNetzbMATHGoogle Scholar
  36. 36.
    Oruç Ö, Bulut F, Esen A (2015) A Haar wavelet-finite difference hybrid method for the numerical solution of the modified Burgers’ equation. J Math Chem.  https://doi.org/10.1007/s10910-015-0507-5 MathSciNetzbMATHGoogle Scholar
  37. 37.
    Aziz I, Siraj-ul-Islam, Šarler B (2013) Wavelet collocation methods for the numerical solution of elliptic BV problems. Appl Math Model 37(3):676–694MathSciNetzbMATHGoogle Scholar
  38. 38.
    Kumar M, Pandit S (2014) A composite numerical scheme for the numerical simulation of coupled Burgers’ equation. Comput Phys Commu 185(3):809–817MathSciNetzbMATHGoogle Scholar
  39. 39.
    Mittal RC, Kaur H, Mishra V (2014) Haar wavelet-based numerical investigation of coupled viscous Burgers’ equation. Int J Comput Math.  https://doi.org/10.1080/00207160.2014.957688 zbMATHGoogle Scholar
  40. 40.
    Kaur H, Mittal RC, Mishra V (2013) Haar wavelet approximate solutions for the generalized Lane–Emden equations arising in astrophysics. Comput Phys Commun 184:2169–2177MathSciNetzbMATHGoogle Scholar
  41. 41.
    Pandit S, Kumar M, Tiwari S (2015) Numerical simulation of second-order hyperbolic telegraph type equations with variable coefficients. Comput Phys Commun 187:83–90MathSciNetzbMATHGoogle Scholar
  42. 42.
    Oruç Ö, Bulut F, Esen A (2016) Numerical solutions of regularized long wave equation by Haar wavelet method. Mediterr J Math 13(5):3235–3253MathSciNetzbMATHGoogle Scholar
  43. 43.
    Oruç Ö, Bulut F, Esen A (2015) A Haar wavelet-finite difference hybrid method for the numerical solution of the modified Burgers’ equation. J Math Chem 53(7):1592–1607MathSciNetzbMATHGoogle Scholar
  44. 44.
    Oruç Ö, Esen A, Bulut F (2016) A Haar wavelet collocation method for coupled nonlinear Schrödinger–KdV equations. Int J Mod Phys C 27(9):1650103.  https://doi.org/10.1142/S0129183116501035 Google Scholar
  45. 45.
    Bulut F, Oruç Ö, Esen A (2015) Numerical solutions of fractional system of partial differential equations by Haar wavelets. Comput Model Eng Sci 108(4):263–284Google Scholar
  46. 46.
    Esen A, Bulut F, Oruç Ö (2016) A unified approach for the numerical solution of time fractional Burgers’ type equations. Eur Phys J Plus 131:116.  https://doi.org/10.1140/epjp/i2016-16116-5 Google Scholar
  47. 47.
    Lepik Ü (2009) Solving fractional integral equations by the Haar wavelet method. Appl Math Comput 214:468–478MathSciNetzbMATHGoogle Scholar
  48. 48.
    Wu JL (2009) A wavelet operational method for solving fractional partial differential equations numerically. Appl Math Comput 214:31–40MathSciNetzbMATHGoogle Scholar
  49. 49.
    Li Y, Zhao W (2010) Haar wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations. Appl Math Comput 216:2276–2285MathSciNetzbMATHGoogle Scholar
  50. 50.
    Rehman M, Ali Khan R (2013) Numerical solutions to initial and boundary value problems for linear fractional partial differential equations. Appl Math Model 37:5233–5244MathSciNetzbMATHGoogle Scholar
  51. 51.
    Ray SS, Patra A (2013) Haar wavelet operational methods for the numerical solutions of fractional order nonlinear oscillatory Van der Pol system. Appl Math Comput 220:659–667MathSciNetzbMATHGoogle Scholar
  52. 52.
    Saeed U, Rehman M (2013) Haar wavelet-quasilinearization technique for fractional nonlinear differential equations. Appl Math Comput 220:630–648MathSciNetzbMATHGoogle Scholar
  53. 53.
    Wang L, Ma Y, Meng Z (2014) Haar wavelet method for solving fractional partial differential equations numerically. Appl Math Comput 227:66–76MathSciNetzbMATHGoogle Scholar
  54. 54.
    Yi M, Huang J (2014) Wavelet operational matrix method for solving fractional differential equations with variable coefficients. Appl Math Comput 230:383–394MathSciNetzbMATHGoogle Scholar
  55. 55.
    Shi Z, Cao Y, Chen QJ (2012) Solving 2D and 3D Poisson equations and biharmonic equations by the Haar wavelet method. Appl Math Model 36:5143–5161MathSciNetzbMATHGoogle Scholar
  56. 56.
    Islam S, Aziz I, Ahmad M (2015) Numerical solution of two-dimensional elliptic PDEs with nonlocal boundary conditions. Comput Math Appl 69:180–205MathSciNetzbMATHGoogle Scholar
  57. 57.
    Aziz I, Siraj-ul-Islam (2013) New algorithms for numerical solution of nonlinear Fredholm and Volterra integral equations using Haar wavelets. J Comput Appl Math 239:333–345MathSciNetzbMATHGoogle Scholar
  58. 58.
    Aziz I, Siraj-ul-Islam, Khan F (2014) A new method based on Haar wavelet for the numerical solution of two-dimensional nonlinear integral equations. J Comput Appl Math 272:70–80MathSciNetzbMATHGoogle Scholar
  59. 59.
    Siraj-ul-Islam, Aziz I, Fayyaz M (2013) A new approach for numerical solution of integro-differential equations via Haar wavelets. Int J Comput Math 90:1971–1989MathSciNetzbMATHGoogle Scholar
  60. 60.
    Siraj-ul-Islam, Aziz I, Al-Fhaid AS (2014) An improved method based on Haar wavelets for numerical solution of nonlinear integral and integro-differential equations of first and higher orders. J Comput Appl Math 260:449–469MathSciNetzbMATHGoogle Scholar
  61. 61.
    Celik I (2013) Haar wavelet approximation for magnetohydrodynamic flow equations. Appl Math Model 37:3894–3902MathSciNetzbMATHGoogle Scholar
  62. 62.
    Shi Z, Yan-Hua X, Jun-ping Z (2016) Haar wavelets method for solving Poisson equations with jump conditions in irregular domain. Adv Comput Math.  https://doi.org/10.1007/s10444-015-9450-z MathSciNetzbMATHGoogle Scholar
  63. 63.
    Samko SG, Kilbas AA, Marichev OI (1993) Fractional integrals and derivatives—theory and applications. Gordon and Breach Science Publishers, PhiladelphiazbMATHGoogle Scholar
  64. 64.
    Zhuang P, Liu F, Anh V, Turner I (2005) Stability and convergence of an implicit numerical method for the nonlinear fractional reaction–subdiffusion process. IMA J Appl Math 74:1–22zbMATHGoogle Scholar
  65. 65.
    Mohammadi F (2015) Haar wavelets approach for solving multidimensional stochastic Itô–Volterra integral equations. Appl Math E Notes 15:80–96MathSciNetzbMATHGoogle Scholar
  66. 66.
    Oliphant TE (2007) Python for scientific computing. Comput Sci Eng 9(3):10–20Google Scholar
  67. 67.
    van der Walt S, Colbert SC, Varoquaux G (2011) The NumPy array: a structure for efficient numerical computation. Comput Sci Eng 13(2):22–30Google Scholar
  68. 68.
    Hunter JD (2007) Matplotlib: a 2D graphics environment. Comput Sci Eng 9(3):90–95Google Scholar
  69. 69.
    Zhang YN, Sun ZZ (2014) Error analysis of a compact ADI scheme for the 2D fractional subdiffusion equation. J Sci Comput 59:104–128MathSciNetzbMATHGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Eğil Vocational and Technical Anatolian High SchoolDiyarbakirTurkey
  2. 2.Department of Mathematics, Faculty of Arts and ScienceInönü UniversityMalatyaTurkey
  3. 3.Department of Physics, Faculty of Arts and ScienceInönü UniversityMalatyaTurkey

Personalised recommendations