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A local meshless method for time fractional nonlinear diffusion wave equation

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Abstract

We present a radial basis function-based local collocation method for solving time fractional nonlinear diffusion wave equation.The main beauty of the local collocation method is that only the nodes located in the subdomain, surrounding the local collocation point, need to be considered when we are calculating the numerical solution at this point. We also prove the unconditional stability and convergence of the proposed scheme. Some numerical experiments are carried out and numerical results are compared with an analytical solution to confirm the efficiency and reliability of the proposed method.

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References

  1. Bagley, R.L., Torvik, P.: A theoretical basis for the application of fractional calculus to viscoelasticity. J. Rheol. 27(3), 201–210 (1983)

    MATH  Google Scholar 

  2. Bhrawy, A., Abdelkawy, M.: A fully spectral collocation approximation for multi-dimensional fractional schrödinger equations. J. Comput. Phys. 294, 462–483 (2015)

    MathSciNet  MATH  Google Scholar 

  3. Cen, Z., Huang, J., Xu, A., Le, A.: Numerical approximation of a time-fractional black–scholes equation. Comput. Math. Appl. 75(8), 2874–2887 (2018)

    MathSciNet  MATH  Google Scholar 

  4. Chandhini, G., Prashanthi, K., Vijesh, V.A.: A radial basis function method for fractional darboux problems. Engineering Analysis with Boundary Elements 86, 1–18 (2018)

    MathSciNet  MATH  Google Scholar 

  5. Chen, W., Ye, L., Sun, H.: Fractional diffusion equations by the kansa method. Comput. Math. Appl. 59(5), 1614–1620 (2010)

    MathSciNet  MATH  Google Scholar 

  6. De Staelen, R., Hendy, A.S.: Numerically pricing double barrier options in a time-fractional black–scholes model. Comput. Math. Appl. 74(6), 1166–1175 (2017)

    MathSciNet  MATH  Google Scholar 

  7. Dehghan, M., Abbaszadeh, M.: Analysis of the element free galerkin (efg) method for solving fractional cable equation with dirichlet boundary condition. Appl. Numer. Math. 109, 208–234 (2016)

    MathSciNet  MATH  Google Scholar 

  8. Dehghan, M., Abbaszadeh, M.: Element free galerkin approach based on the reproducing kernel particle method for solving 2d fractional tricomi-type equation with robin boundary condition. Comput. Math. Appl. 73(6), 1270–1285 (2017)

    MathSciNet  MATH  Google Scholar 

  9. Dehghan, M., Abbaszadeh, M.: Two meshless procedures: moving kriging interpolation and element-free galerkin for fractional pdes. Appl. Anal. 96(6), 936–969 (2017)

    MathSciNet  MATH  Google Scholar 

  10. Dehghan, M., Abbaszadeh, M.: The use of proper orthogonal decomposition (pod) meshless rbf-fd technique to simulate the shallow water equations. J. Comput. Phys. 351, 478–510 (2017)

    MathSciNet  MATH  Google Scholar 

  11. Dehghan, M., Abbaszadeh, M.: An efficient technique based on finite difference/finite element method for solution of two-dimensional space/multi-time fractional bloch–torrey equations. Appl. Numer. Math. 131, 190–206 (2018)

    MathSciNet  MATH  Google Scholar 

  12. Dehghan, M., Abbaszadeh, M., Mohebbi, A.: An implicit rbf meshless approach for solving the time fractional nonlinear sine-gordon and klein–gordon equations. Engineering Analysis with Boundary Elements 50, 412–434 (2015)

    MathSciNet  MATH  Google Scholar 

  13. Dehghan, M., Abbaszadeh, M., Mohebbi, A.: A meshless technique based on the local radial basis functions collocation method for solving parabolic–parabolic patlak–keller–segel chemotaxis model. Engineering Analysis with Boundary Elements 56, 129–144 (2015)

    MathSciNet  MATH  Google Scholar 

  14. Dehghan, M., Abbaszadeh, M., Mohebbi, A.: Analysis of a meshless method for the time fractional diffusion-wave equation. Numerical Algorithms 73(2), 445–476 (2016)

    MathSciNet  MATH  Google Scholar 

  15. Dehghan, M., Manafian, J., Saadatmandi, A.: Solving nonlinear fractional partial differential equations using the homotopy analysis method. Numerical Methods for Partial Differential Equations: An International Journal 26(2), 448–479 (2010)

    MathSciNet  MATH  Google Scholar 

  16. Dehghan, M., Mohammadi, V.: Two-dimensional simulation of the damped kuramoto–sivashinsky equation via radial basis function-generated finite difference scheme combined with an exponential time discretization. Engineering Analysis with Boundary Elements 107, 168–184 (2019)

    MathSciNet  MATH  Google Scholar 

  17. 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)

    MathSciNet  MATH  Google Scholar 

  18. Dehghan, M., Shokri, A.: A numerical method for solution of the two-dimensional sine-gordon equation using the radial basis functions. Math. Comput. Simul. 79(3), 700–715 (2008)

    MathSciNet  MATH  Google Scholar 

  19. Gao, G.H., Sun, Z.Z., Zhang, Y.N.: A finite difference scheme for fractional sub-diffusion equations on an unbounded domain using artificial boundary conditions. J. Comput. Phys. 231(7), 2865–2879 (2012)

    MathSciNet  MATH  Google Scholar 

  20. Ghehsareh, H.R., Bateni, S.H., Zaghian, A.: A meshfree method based on the radial basis functions for solution of two-dimensional fractional evolution equation. Engineering Analysis with Boundary Elements 61, 52–60 (2015)

    MathSciNet  MATH  Google Scholar 

  21. Ghehsareh, H.R., Zaghian, A., Raei, M.: A local weak form meshless method to simulate a variable order time-fractional mobile–immobile transport model. Engineering Analysis with Boundary Elements 90, 63–75 (2018)

    MathSciNet  MATH  Google Scholar 

  22. Golbabai, A., Nikpour, A.: Computing a numerical solution of two dimensional non-linear schrödinger equation on complexly shaped domains by rbf based differential quadrature method. J. Comput. Phys. 322, 586–602 (2016)

    MathSciNet  MATH  Google Scholar 

  23. Gu, Y., Zhuang, P.: Anomalous sub-diffusion equations by the meshless collocation method. Aust. J. Mech. Eng. 10(1), 1–8 (2012)

    Google Scholar 

  24. Gu, Y., Zhuang, P., Liu, Q.: An advanced meshless method for time fractional diffusion equation. Int. J. Comput. Methods 8(04), 653–665 (2011)

    MathSciNet  MATH  Google Scholar 

  25. Hosseini, V.R., Chen, W., Avazzadeh, Z.: Numerical solution of fractional telegraph equation by using radial basis functions. Engineering Analysis with Boundary Elements 38, 31–39 (2014)

    MathSciNet  MATH  Google Scholar 

  26. Hosseini, V.R., Shivanian, E., Chen, W.: Local radial point interpolation (mlrpi) method for solving time fractional diffusion-wave equation with damping. J. Comput. Phys. 312, 307–332 (2016)

    MathSciNet  MATH  Google Scholar 

  27. Jin, B., Lazarov, R., Liu, Y., Zhou, Z.: The galerkin finite element method for a multi-term time-fractional diffusion equation. J. Comput. Phys. 281, 825–843 (2015)

    MathSciNet  MATH  Google Scholar 

  28. Kumar, A., Bhardwaj, A., Kumar, B.V.R.: A meshless local collocation method for time fractional diffusion wave equation. Comput. Math. Appl. 78(6), 1851–1861 (2019)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  30. Mardani, A., Hooshmandasl, M., Heydari, M., Cattani, C.: A meshless method for solving the time fractional advection–diffusion equation with variable coefficients. Comput. Math. Appl. 75(1), 122–133 (2018)

    MathSciNet  MATH  Google Scholar 

  31. Mohebbi, A., Abbaszadeh, M., Dehghan, M.: 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. Engineering Analysis with Boundary Elements 37(2), 475–485 (2013)

    MathSciNet  MATH  Google Scholar 

  32. Mohebbi, A., Abbaszadeh, M., Dehghan, M.: Solution of two-dimensional modified anomalous fractional sub-diffusion equation via radial basis functions (rbf) meshless method. Engineering Analysis with Boundary Elements 38, 72–82 (2014)

    MathSciNet  MATH  Google Scholar 

  33. Nagy, A.: Numerical solution of time fractional nonlinear klein–gordon equation using sinc–chebyshev collocation method. Appl. Math. Comput. 310, 139–148 (2017)

    MathSciNet  MATH  Google Scholar 

  34. Quarteroni, A., Valli, A.: Numerical approximation of partial differential equations, vol. 23. Springer Science & Business Media (2008)

  35. Salehi, R.: A meshless point collocation method for 2-d multi-term time fractional diffusion-wave equation. Numerical Algorithms 74(4), 1145–1168 (2017)

    MathSciNet  MATH  Google Scholar 

  36. Sarra, S.A.: A local radial basis function method for advection–diffusion–reaction equations on complexly shaped domains. Appl. Math. Comput. 218(19), 9853–9865 (2012)

    MathSciNet  MATH  Google Scholar 

  37. Sun, H., Liu, X., Zhang, Y., Pang, G., Garrard, R.: A fast semi-discrete kansa method to solve the two-dimensional spatiotemporal fractional diffusion equation. J. Comput. Phys. 345, 74–90 (2017)

    MathSciNet  MATH  Google Scholar 

  38. Sun, Z.Z., Wu, X.: A fully discrete difference scheme for a diffusion-wave system. Appl. Numer. Math. 56(2), 193–209 (2006)

    MathSciNet  MATH  Google Scholar 

  39. Tayebi, A., Shekari, Y., Heydari, M.: A meshless method for solving two-dimensional variable-order time fractional advection–diffusion equation. J. Comput. Phys. 340, 655–669 (2017)

    MathSciNet  MATH  Google Scholar 

  40. Uddin, M., Haq, S.: Rbfs approximation method for time fractional partial differential equations. Commun. Nonlinear Sci. Numer. Simul. 16(11), 4208–4214 (2011)

    MathSciNet  MATH  Google Scholar 

  41. Vong, S., Wang, Z.: A compact difference scheme for a two dimensional fractional klein–gordon equation with neumann boundary conditions. J. Comput. Phys. 274, 268–282 (2014)

    MathSciNet  MATH  Google Scholar 

  42. Yan, L., Yang, F.: Efficient kansa-type mfs algorithm for time-fractional inverse diffusion problems. Comput. Math. Appl.s 67(8), 1507–1520 (2014)

    MathSciNet  MATH  Google Scholar 

  43. Yang, J., Zhao, Y., Liu, N., Bu, W., Xu, T., Tang, Y.: An implicit mls meshless method for 2-d time dependent fractional diffusion–wave equation. Appl. Math. Model. 39(3-4), 1229–1240 (2015)

    MathSciNet  MATH  Google Scholar 

  44. Zeng, F., Zhang, Z., Karniadakis, G.E.: Fast difference schemes for solving high-dimensional time-fractional subdiffusion equations. J. Comput. Phys. 307, 15–33 (2016)

    MathSciNet  MATH  Google Scholar 

  45. Zhuang, P., Gu, Y., Liu, F., Turner, I., Yarlagadda, P.: Time-dependent fractional advection–diffusion equations by an implicit mls meshless method. Int. J. Numer. Methods Eng. 88(13), 1346–1362 (2011)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

We would like to thank reviewers for their comments and suggestions that really improved the quality of the paper.

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  1. Department of Basic Sciences and Humanities, Rajiv Gandhi Institute of Petroleum Technology, Amethi, 229304, India

    Alpesh Kumar & Akanksha Bhardwaj

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  1. Alpesh Kumar
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Correspondence to Akanksha Bhardwaj.

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Kumar, A., Bhardwaj, A. A local meshless method for time fractional nonlinear diffusion wave equation. Numer Algor 85, 1311–1334 (2020). https://doi.org/10.1007/s11075-019-00866-9

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