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An improved meshless method for solving two-dimensional distributed order time-fractional diffusion-wave equation with error estimate

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Abstract

In the current decade, the meshless methods have been developed for solving partial differential equations. The meshless methods may be classified in two basic parts:

  1. 1.

    The meshless methods based on the strong form

  2. 2.

    The meshless methods based on the weak form

The element-free Galerkin (EFG) method is a meshless method based on the global weak form. The test and trial functions in element-free Galerkin are shape functions of moving least squares (MLS) approximation. Also, the traditional MLS shape functions have not the δ-Kronecker property. Recently, a new class of MLS shape functions has been presented. These are well-known as the interpolating MLS (IMLS) shape functions. The IMLS shape functions have the δ-Kronecker property; thus the essential boundary conditions can be applied directly. The main aim of this paper is to combine the alternating direction implicit approach with the IEFG method. To this end, we apply the mentioned technique on the distributed order time-fractional diffusion-wave equation. For comparing the numerical results, we propose three schemes based on the trapezoidal, Simpson, and Gauss-Legendre quadrature techniques. Also, we investigate the uniqueness, existence and stability analysis of the new schemes and we obtain an error estimate for the full-discrete schemes. The time-fractional derivative has been described in Caputo’s sense. Numerical examples demonstrate the theoretical results and the efficiency of the proposed schemes.

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References

  1. Atanackovic, T. M., Pilipovic, S., Zorica, D.: Time distributed-order diffusion-wave equation. II. Applications of Laplace and Fourier transformations. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science 465, 1893–1917 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  2. Atanackovic, T. M., Pilipovic, S., Zorica, D.: Existence and calculation of the solution to the time distributed order diffusion equation. Phys. Scr. 2009, 014012 (2009)

    Article  Google Scholar 

  3. Atkinson, K. E.: An Introduction to Numerical Analysis. Wiley, New York (1989)

    MATH  Google Scholar 

  4. Bagley, R., Torvik, P.: A theoretical basis for the application of fractional calculus to viscoelasticity. J. Rheol. 27, 201–210 (1983)

    Article  MATH  Google Scholar 

  5. Belytschko, T., Lu, Y. Y., Gu, L.: Element free Galerkin methods. Int. J. Numer. Methods Eng. 37, 229–256 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  6. Belytschko, T., Krongauz, Y., Organ, D., Fleming, M., Krysl, P.: Meshless methods: an overview and recent developments. Comput. Methods Appl. Mech. Engrg. 139, 3–47 (1996)

    Article  MATH  Google Scholar 

  7. Browder, F. E.: Existence and uniqueness theorems for solutions of nonlinear boundary value problems. In: Finn, R. (ed.) Applications of Nonlinear P.D. Es, Proceedings of Symposium of Applied Mathematics, vol. 17, pp 24–49 (1965)

  8. Chechkin, A. V., Gorenflo, R., Sokolov, I. M., Gonchar, V. Y.: Distributed order time fractional diffusion equation. Fract. Calc. Appl. Anal. 6, 259–280 (2003)

    MathSciNet  MATH  Google Scholar 

  9. Chechkin, A. V., Gorenflo, R., Sokolov, I. M.: Retarding subdiffusion and accelerating superdiffusion governed by distributed-order fractional diffusion equations. Phys. Rev. E. 66, 046129 (2002)

    Article  Google Scholar 

  10. Chen, W., Liang, Y., Hu, S., Sun, H.: Fractional derivative anomalous diffusion equation modeling prime number distribution. Fractional Calculus and Applied Analysis 18, 789–798 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  11. Cheng, R., Cheng, Y.: Error estimates for the finite point method. Appl. Numer. Math. 58, 884–898 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  12. Cheng, Y., Bai, F. N., Peng, M. J.: A novel interpolating element free Galerkin (IEFG) method for two-dimensional elastoplasticity. Appl. Math. Model. 38, 5187–5197 (2014)

    Article  MathSciNet  Google Scholar 

  13. Cheng, Y., Peng, M.: Boundary element free method for elastodynamics. Science in China G 48, 641–657 (2005)

    Article  Google Scholar 

  14. Chung, H. J., Belytschko, T.: An error estimate in the EFG method. Comput. Mech. 21, 91–100 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  15. Cui, M.: Compact alternating direction implicit method for two-dimensional time fractional diffusion equation. J. Comput. Phy. 231, 2621–2633 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  16. Cui, M.: Convergence analysis of high-order compact alternating direction implicit schemes for the two-dimensional time fractional diffusion equation. Numer. Algor. 62, 383–409 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  17. Diethelm, K., Ford, N. J.: Analysis of fractional differential equations. J. Math. Anal. Appl. 265, 229–248 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  18. Diethelm, K., Ford, N. J.: Numerical analysis for distributed-order differential equations. J. Comput. Appl. Math. 225, 96–104 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  19. Dehghan, M., Mohebbi, A.: High-order compact boundary value method for the solution of unsteady convection-diffusion problems. Math. Comput. Simul. 79, 683–699 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  20. Dehghan, M., Safarpoor, M., Abbaszadeh, M.: Two high-order numerical algorithms for solving the multi-term time fractional diffusion-wave equations. J. Comput. Appl. Math. 290, 174–195 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  21. Dehghan, M., Abbaszadeh, M., Mohebbi, A.: Error estimate for the numerical solution of fractional reaction-subdiffusion process based on a meshless method. J. Comput. Appl. Math. 280, 14–36 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  22. Dehghan, M., Salehi, R.: A meshless local Petrov–Galerkin method for the time-dependent Maxwell equations. J. Comput. Appl. Math. 268, 93–110 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  23. Dehghan, M., Salehi, R.: The numerical solution of the non-linear integro-differential equations based on the meshless method. J. Comput. Appl. Math. 236, 2367–2377 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  24. Dehghan, M., Manafian, J., Saadatmandi, A.: Solving nonlinear fractional partial differential equations using the homotopy analysis method. Numerical Methods for Partial Differential Equations 26, 448–479 (2010)

    MathSciNet  MATH  Google Scholar 

  25. Dehghan, M.: A new ADI technique for two-dimensional parabolic equation with an integral condition. Comput. Math. Appl. 43, 1477–1488 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  26. Fu, Z. J., Chen, W., Yang, H. T.: Boundary particle method for Laplace transformed time fractional diffusion equations. J. Comput. Phys. 235, 52–66 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  27. Gao, G. H., Sun, Z. Z.: Two alternating direction implicit difference schemes with the extrapolation method for the two-dimensional distributed-order differential equations. Comput. Math. Appl. 69, 926–948 (2015)

    Article  MathSciNet  Google Scholar 

  28. Katsikadelis, J. T.: Numerical solution of distributed order fractional differential equations. J. Comput. Phys. 259, 11–22 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  29. Krongauz, Y., Belytschko, T.: EFG approximation with discontinuous derivatives. Int. J. Numer. Meth. Eng. 41, 1215–1233 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  30. Hu, X., Liu, F., Turner, I., Anh, V.: An implicit numerical method of a new time distributed-order and two-sided space-fractional advection-dispersion equation. Numerical Algorithm 72, 393–407 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  31. Lancaster, P., Salkauskas, K.: Surfaces generated by moving least squares methods. Math. Comput. 37, 141–158 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  32. Lee, C. K., Zhou, C. E.: On error estimation and adaptive refinement for element free Galerkin method part I: stress recovery and a posteriori error estimation. Comput. Struct. 82, 413–428 (2004)

    Article  Google Scholar 

  33. Lee, C. K., Zhou, C. E.: On error estimation and adaptive refinement for element free Galerkin method part II: adaptive refinement. Comput. Struct. 82, 429–443 (2004)

    Article  Google Scholar 

  34. Li, L., Xu, D., Luo, M.: Alternating direction implicit Galerkin finite element method for the two-dimensional fractional diffusion-wave equation. J. Comput. Phys. 255, 471–485 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  35. Li, C. P., Zhao, Z. G., Chen, Y. Q.: Numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion. Comput. Math. Appl. 62, 855–875 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  36. Li, C. P., Chen, A., Ye, J.: Numerical approaches to fractional calculus and fractional ordinary differential equation. J. Comput. Phys. 230, 3352–3368 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  37. Li, C. P., Deng, W.: Remarks on fractional derivatives. Appl. Math. Comput. 187, 777–784 (2007)

    MathSciNet  MATH  Google Scholar 

  38. Li, C. P., Dao, X., Guo, P.: Fractional derivatives in complex planes. Nonlinear Anal. Theory Methods Appl. 71, 1857–1869 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  39. Li, C. P., Yan, J. P.: The synchronization of three fractional differential systems. Chaos, Solitons Fractals 32, 751–757 (2007)

    Article  Google Scholar 

  40. Liao, H. L., Zhang, Y. N., Zhao, Y., Shi, H. S.: Stability and convergence of modified Du Fort-Frankel schemes for solving time-fractional subdiffusion equations. J. Sci. Comput. 61, 629–648 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  41. Liu, F., Yang, C., Burrage, K.: Numerical method and analytical technique of the modified anomalous subdiffusion equation with a nonlinear source term. J. Comput. Appl Math. 231, 160–176 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  42. Liu, F., Zhuang, P., Turner, I., Burrage, K., Anh, V.: A new fractional finite volume method for solving the fractional diffusion equation. Appl. Math. Model. 38, 3871–3878 (2014)

    Article  MathSciNet  Google Scholar 

  43. Liu, F., Zhuang, P., Turner, I., Anh, V., Burrage, K.: A semi-alternating direction method for a 2-D fractional FitzHugh-Nagumo monodomain model on an approximate irregular domain. J. Comput. Phys. 293, 252–263 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  44. Liu, Q., Liu, F., Gu, Y., Zhuang, P., Chen, J., Turner, I.: A meshless method based on point interpolation method (PIM) for the space fractional diffusion equation. Appl. Math. Comp. 256, 930–938 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  45. Liu, Q., Liu, F., Turner, I., Anh, V., Gu, Y.: A RBF meshless approach for modeling a fractal mobile/immobile transport model. Appl. Math. Comp. 226, 336–347 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  46. Luchko, Y.: Boundary value problems for the generalized time-fractional diffusion equation of distributed order. Fract. Calc. Appl. Anal. 12, 409–422 (2009)

    MathSciNet  MATH  Google Scholar 

  47. Mainardi, F., Pagnini, G., Gorenflo, R.: Some aspects of fractional diffusion equations of single and distributed order. Appl. Math. Comput. 187, 295–305 (2007)

    MathSciNet  MATH  Google Scholar 

  48. Mainardi, F., Mura, A., Pagnini, G., Gorenflo, R.: Time-fractional diffusion of distributed order. J. Vib. Control 14, 1267–1290 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  49. Meerschaert, M. M., Nane, E., Vellaisamy, P.: Distributed-order fractional diffusions on bounded domains. J. Math. Anal. Appl. 379, 216–228 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  50. Metzler, R., Klafter, J.: The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics. J. Phys. A 37, R161–208 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  51. Oldham, K. B., Spanier, J.: The Fractional Calculus: Theory and Application of Differentiation and Integration to Arbitrary Order. Academic Press (1974)

  52. Peng, M. J., Li, R. X., Cheng, Y. M.: Analyzing three-dimensional viscoelasticity problems via the improved element-free Galerkin (IEFG) method. Eng. Anal. Bound. Elemen. 40, 104–113 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  53. Podulbny, I.: Fractional Differential Equations. Academic Press, New York (1999)

    Google Scholar 

  54. Podlubny, I., Skovranek, T., Jara, B. M. V., Petras, I., Verbitsky, V., Chen, Y.: Matrix approach to discrete fractional calculus III: nonequidistant grids, variable step length and distributed orders. Philos. Trans. R. Soc. A Math. Phys. Eng. Sci., 371 (2013)

  55. Quarteroni, A., Valli, A.: Numerical Approximation of Partial Differential Equations. Springer, New York (1997)

    MATH  Google Scholar 

  56. Ren, H., Cheng, Y.: The interpolating element-free Galerkin (IEFG) method for two-dimensional potential problems. Eng. Anal. Boundary Elem. 36, 873–880 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  57. Sandev, T., Chechkin, A. V., Korabel, N., Kantz, H., Sokolov, I. M., Metzler, R.: Distributed-order diffusion equations and multifractality: models and solutions. Phys. Rev. E 92, 042117 (2015)

    Article  MathSciNet  Google Scholar 

  58. Sun, Z. Z., Wu, X. N.: A fully discrete difference scheme for a diffusion-wave system. Appl. Numer. Math. 56, 193–209 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  59. Sun, H., Chen, W., Li, C. P., Chen, Y.: Finite difference schemes for variable-order time fractional diffusion equation. Int. J. Bifurcation Chaos 22(4), 1250085 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  60. Wei, S., Chen, W., Hon, Y. C.: Implicit local radial basis function method for solving two-dimensional time fractional diffusion equations. Therm. Sci. 19(1), S59–S67 (2015)

    Article  Google Scholar 

  61. Wess, W.: The fractional diffusion equation. J. Math. Phys. 27, 2782–2785 (1996)

    Article  MathSciNet  Google Scholar 

  62. Ye, H., Liu, F., Anh, V.: Compact difference scheme for distributed-order time-fractional diffusion-wave equation on bounded domains. J. Comput. Phys. 298, 652–660 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  63. Ye, H., Liu, F., Anh, V., Turner, I.: Numerical analysis for the time distributed order and Riesz space fractional diffusions on bounded domains. IMA J. Appl. Math. 80(3), 531–540 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  64. Yuste, S. B.: Fast, accurate and robust adaptive finite difference methods for fractional diffusion equations. Numerical Algorithms 71, 207–228 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  65. Yuste, S. B., Quintana-Murillo, J.: A finite difference method with non-uniform timesteps for fractional diffusion equations. Comput. Phys. Commun. 183, 2594–2600 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  66. Yuste, S. B.: Weighted average finite difference methods for fractional diffusion, equations. J. Comput. Phy. 216, 264–274 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  67. Zeng, F., Li, C. P., Liu, F., Turner, I.: Numerical algorithms for time-fractional subdiffusion equation with second-order accuracy. SIAM J. Sci. Comput. 37, A55–A78 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  68. Zeng, F.: Second-order stable finite difference schemes for the time-fractional diffusion-wave equation. J. Sci. Comput. 65, 1–20 (2014)

    MathSciNet  Google Scholar 

  69. Zeng, F., Li, C. P., Liu, F., Turner, I.: The use of finite difference/element approaches for solving the time-fractional subdiffusion equation. SIAM J. Sci. Comput. 35, A2976–A3000 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  70. Zeng, F., Liu, F., Li, C., Burrage, K., Turner, I., Anh, V.: Crank-Nicolson ADI spectral method for the two-dimensional Riesz space fractional nonlinear reaction-diffusion equation. SIAM J. Numer. Anal. 52-6, 2599–2622 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  71. Zhang, L. W., Deng, Y. J., Liew, K. M., Cheng, Y. M.: The improved complex variable element free Galerkin method for two-dimensional Schrödinger equation. Comput. Math. Appl. 68, 1093–1106 (2014)

    Article  MathSciNet  Google Scholar 

  72. Zhang, L. W., Li, D. M., Liew, K. M.: An element-free computational framework for elastodynamic problems based on the IMLS-Ritz method. Eng. Anal. Bound. Elem. 54, 39–46 (2015)

    Article  MathSciNet  Google Scholar 

  73. Zhang, L. W., Lei, Z. X., Liew, K. M.: Buckling analysis of FG-CNT reinforced composite thick skew plates using an element-free approach. Composites Part B: Engineering 75, 36–46 (2015)

    Article  Google Scholar 

  74. Zhang, L. W., Lei, Z. X., Liew, K. M.: Free vibration analysis of functionally graded carbon nanotube-reinforced composite triangular plates using the FSDT and element-free IMLS-Ritz method. Compos. Struct. 120, 189–199 (2015)

    Article  Google Scholar 

  75. Zhang, L. W., Deng, Y. J., Liew, K. M.: An improved element-free Galerkin method for numerical modeling of the biological population problems. Eng. Anal. Bound. Elem. 40, 181–188 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  76. Zhang, Z., Hao, S. Y., Liew, K. M., Cheng, Y. M.: The improved element-free Galerkin method for two-dimensional elastodynamics problems. Eng. Anal. Bound. Elem. 37, 1576–1584 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  77. Zhang, Y. N., Sun, Z. Z., Wu, H. W.: Error estimate of Crank-Nicolson-tape difference schemes for the subdiffusion equation. SIAM J. Numer. Anal. 49, 2302–2322 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  78. Zhang, Y. N., Sun, Z. Z., Zhao, X.: Compact alternating direction implicit scheme for the two-dimensional fractional diffusion-wave equation. SIAM J. Numer. Anal. 50, 1535–1555 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  79. Zhang, Z., Liew, K. M., Cheng, Y.: Coupling of the improved element-free Galerkin and boundary element methods for two-dimensional elasticity problems. Eng. Anal. Bound. Elem. 32, 100–107 (2008)

    Article  MATH  Google Scholar 

  80. Zhao, X., Sun, Z. Z.: Compact Crank-Nicolson schemes for a class of fractional Cattaneo equation in inhomogeneous medium. J. Sci. Comput. 62, 1–25 (2014)

    MathSciNet  Google Scholar 

  81. Zhao, X., Xu, Q.: Efficient numerical schemes for fractional sub-diffusion equation with the spatially variable coefficient. Appl. Math. Model. 38, 3848–3859 (2014)

    Article  MathSciNet  Google Scholar 

  82. Zhuang, P., Liu, F., Turner, I., Gu, Y. T.: Finite volume and finite element methods for solving a one-dimensional space-fractional Boussinesq equation. Appl. Math Model. 38, 3860–3870 (2014)

    Article  MathSciNet  Google Scholar 

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  1. Department of Applied Mathematics, Faculty of Mathematics and Computer Sciences, Amirkabir University of Technology, No. 424, Hafez Ave., 15914, Tehran, Iran

    Mostafa Abbaszadeh & Mehdi Dehghan

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  1. Mostafa Abbaszadeh
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  2. Mehdi Dehghan
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Abbaszadeh, M., Dehghan, M. An improved meshless method for solving two-dimensional distributed order time-fractional diffusion-wave equation with error estimate. Numer Algor 75, 173–211 (2017). https://doi.org/10.1007/s11075-016-0201-0

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