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An efficient differential quadrature method for fractional advection–diffusion equation

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Abstract

This article studies a direct numerical approach for fractional advection–diffusion equations (ADEs). Using a set of cubic trigonometric B-splines as test functions, a differential quadrature (DQ) method is firstly proposed for the 1D and 2D time-fractional ADEs of order (0, 1]. The weighted coefficients are determined, and with them, the original equation is transformed into a group of general ordinary differential equations (ODEs), which are discretized by an effective difference scheme or Runge–Kutta method. The stability is investigated under a mild theoretical condition. Secondly, based on a set of cubic B-splines, we develop a new Crank–Nicolson type DQ method for the 2D space-fractional ADEs without advection. The DQ approximations to fractional derivatives are introduced, and the values of the fractional derivatives of B-splines are computed by deriving explicit formulas. The presented DQ methods are evaluated on five benchmark problems and the simulations of the unsteady propagation of solitons and Gaussian pulse. In comparison with the algorithms in the open literature, numerical results finally illustrate the validity and accuracy.

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References

  1. Abbas, M., Majid, A.A., Ismail, A.I.M., Rashid, A.: The application of cubic trigonometric B-spline to the numerical solution of the hyperbolicproblems. Appl. Math. Comput. 239, 74–88 (2014)

    MathSciNet  MATH  Google Scholar 

  2. Bellman, R., Casti, J.: Differential quadrature and long-term integration. J. Math. Anal. Appl. 34(2), 235–238 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bhrawy, A., Zaky, M.: A fractional-order Jacobi Tau method for a class of time-fractional PDEs with variable coefficients. Math. Method Appl. Sci. 39(7), 1765–1779 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chen, M.H., Deng, W.H.: Fourth order difference approximations for space Riemann–Liouville derivatives based on weighted and shifted Lubich difference operators. Commun. Comput. Phys. 16(2), 516–540 (2014)

    Article  MathSciNet  Google Scholar 

  5. Chen, Y.M., Wu, Y.B., Cui, Y.H., Wang, Z.Z., Jin, D.M.: Wavelet method for a class of fractional convection–diffusion equation with variable coefficients. J. Comput. Sci. 1(3), 146–149 (2010)

    Article  Google Scholar 

  6. Cui, M.R.: A high-order compact exponential scheme for the fractional convection–diffusion equation. J. Comput. Appl. Math. 255, 404–416 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  7. Dehghan, M., Mohebbi, A.: High-order compact boundary value method for the solution of unsteady convection–diffusion problems. Math. Comput. Simulat. 79(3), 683–699 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. Douglas Jr., J., Russell, T.F.: Numerical methods for convection–dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures. SIAM J. Numer. Anal. 19(5), 871–885 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  9. Esmaeili, S., Garrappa, R.: A pseudo-spectral scheme for the approximate solution of a time-fractional diffusion equation. Int. J. Comput. Math. 92(5), 980–994 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  10. Gao, G.H., Sun, H.W.: Three-point combined compact alternating direction implicit difference schemes for two-dimensional time-fractional advection–diffusion equations. Commun. Comput. Phys. 17(2), 487–509 (2015)

    Article  MathSciNet  Google Scholar 

  11. Heydari, M.H.: Wavelets Galerkin method for the fractional subdiffusion equation. J. Comput. Nonlinear Dyn. 11(6), 061014 (2016)

    Article  Google Scholar 

  12. Hosseini, S.M., Ghaffari, R.: Polynomial and nonpolynomial spline methods for fractional sub-diffusion equations. Appl. Math. Model. 38(14), 3554–3566 (2014)

    Article  MathSciNet  Google Scholar 

  13. Huang, C.B., Yu, X.J., Wang, C., Li, Z.Z., An, N.: A numerical method based on fully discrete direct discontinuous Galerkin method for the time fractional diffusion equation. Appl. Math. Comput. 264, 483–492 (2015)

    MathSciNet  Google Scholar 

  14. Izadkhah, M.M., Saberi-Nadjafi, J.: Gegenbauer spectral method for time-fractional convection–diffusion equations with variable coefficients. Math. Method Appl. Sci. 38(15), 3183–3194 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  15. Jia, J.H., Wang, H.: A fast finite volume method for conservative space-fractional diffusion equations in convex domains. J. Comput. Phys. 310, 63–84 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  16. Ji, C.C., Sun, Z.Z.: A high-order compact finite difference scheme for the fractional sub-diffusion equation. J. Sci. Comput. 64(3), 959–985 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  17. Jiang, Y.J., Ma, J.T.: High-order finite element methods for time-fractional partial differential equations. J. Comput. Appl. Math. 235(11), 3285–3290 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. Jin, B.T., Lazarov, R., Zhou, Z.: Error estimates for a semidiscrete finite element method for fractional order parabolic equations. SIAM J. Numer. Anal. 51(1), 445–466 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  19. Kalita, J.C., Dalal, D.C., Dass, A.K.: A class of higher order compact schemes for the unsteady two-dimensional convection–diffusion equation with variable convection coefficients. Int. J. Numer. Methods Fluids 38(12), 1111–1131 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  20. Karaa, S., Zhang, J.: High order ADI method for solving unsteady convection–diffusion problems. J. Comput. Phys. 198(1), 1–9 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  21. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)

    MATH  Google Scholar 

  22. Laub, A.J.: Matrix Analysis For Scientists & Engineers. Society for Industrial and Applied Mathematics, Philadelphia (2005)

    Book  MATH  Google Scholar 

  23. Lin, Y.M., Xu, C.J.: Finite difference/spectral approximations for the time-fractional diffusion equation. J. Comput. Phys. 225(2), 1533–1552 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  24. Liu, Q., Gu, Y.T., Zhuang, P., Liu, F., Nie, Y.F.: An implicit RBF meshless approach for time fractional diffusion equations. Comput. Mech. 48(1), 1–12 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  25. Luo, W.H., Huang, T.Z., Wu, G.C., Gu, X.M.: Quadratic spline collocation method for the time fractional subdiffusion equation. Appl. Math. Comput. 276, 252–265 (2016)

    MathSciNet  Google Scholar 

  26. Mittal, R.C., Jain, R.K.: Numerical solutions of nonlinear Burgers’ equation with modified cubic B-splines collocation method. Appl. Math. Comput. 218(15), 7839–7855 (2012)

    MathSciNet  MATH  Google Scholar 

  27. Ma, Y.B., Sun, C.P., Haake, D.A., Churchill, B.M., Ho, C.M.: A high-order alternating direction implicit method for the unsteady convection-dominated diffusion problem. Int. J. Numer. Methods Fluids 70(6), 703–712 (2012)

    Article  MathSciNet  Google Scholar 

  28. Meerschaert, M.M., Scheffler, H.P., Tadjeran, C.: Finite difference methods for two-dimensional fractional dispersion equation. J. Comput. Phys. 211(1), 249–261 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  29. Mainardi, F.: Fractals and Fractional Calculus Continuum Mechanics. Springer, Berlin (1997)

    MATH  Google Scholar 

  30. Nazir, T., Abbas, M., Ismail, A.I.M., Majid, A.A., Rashid, A.: The numerical solution of advection–diffusion problems using new cubic trigonometric B-splines approach. Appl. Math. Model. 40(7–8), 4586–4611 (2016)

    Article  MathSciNet  Google Scholar 

  31. Noye, B.J., Tan, H.H.: Finite difference methods for solving the two-dimensional advection–diffusion equation. Int. J. Numer. Methods Fluids 9(1), 75–98 (1989)

    Article  MATH  Google Scholar 

  32. Peaceman, D.W., Rachford Jr., H.H.: The numerical solution of parabolic and elliptic differential equations. J. Soc. Ind. Appl. Math. 3(1), 28–41 (1955)

    Article  MathSciNet  MATH  Google Scholar 

  33. Pirkhedri, A., Javadi, H.H.S.: Solving the time-fractional diffusion equation via Sinc–Haar collocation method. Appl. Math. Comput. 257, 317–326 (2015)

    MathSciNet  MATH  Google Scholar 

  34. Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)

    MATH  Google Scholar 

  35. Razminia, K., Razminia, A., Baleanu, D.: Investigation of the fractional diffusion equation based on generalized integral quadrature technique. Appl. Math. Model. 39(1), 86–98 (2015)

    Article  MathSciNet  Google Scholar 

  36. Roop, J.P.: Computational aspects of FEM approximation of fractional advection dispersion equations on bounded domains in \({\mathbb{R}}^2\). J. Comput. Appl. Math. 193(1), 243–268 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  37. Saadatmandi, A., Dehghan, M., Azizi, M.R.: The Sinc–Legendre collocation method for a class of fractional convection–diffusion equations with variable coefficients. Commun. Nonlinear Sci. Numer. Simul. 17(11), 4125–4136 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  38. Sayevand, K., Yazdani, A., Arjang, F.: Cubic B-spline collocation method and its application for anomalous fractional diffusion equations in transport dynamic systems. J. Vib. Control 22(9), 2173–2186 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  39. Shirzadi, A., Ling, L., Abbasbandy, S.: Meshless simulations of the two-dimensional fractional-time convection–diffusion–reaction equations. Eng. Anal. Bound. Elem. 36(11), 1522–1527 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  40. Shlesinger, M.F., West, B.J., Klafter, J.: Lévy dynamics of enhanced diffusion: application to turbulence. Phys. Rev. Lett. 58(11), 1100–1103 (1987)

    Article  MathSciNet  Google Scholar 

  41. Shu, C., Richards, B.E.: Application of generalized differential quadrature to solve two-dimensional incompressible Navier–Stokes equations. Int. J. Numer. Methods Fluids 15(7), 791–798 (1992)

    Article  MATH  Google Scholar 

  42. Tomasiello, S.: Stability and accuracy of the iterative differential quadrature method. Int. J. Numer. Methods Eng. 58(9), 1277–1296 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  43. Tomasiello, S.: Numerical stability of DQ solutions of wave problems. Numer. Algor. 57(3), 289–312 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  44. Tian, W.Y., Zhou, H., Deng, W.H.: A class of second order difference approximations for solving space fractional diffusion equations. Math. Comput. 84(294), 1703–1727 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  45. Thai-Quang, N., Mai-Duy, N., Tran, C.-D., Tran-Cong, T.: High-order alternating direction implicit method based on compact integrated-RBF approximations for unsteady/steady convection–diffusion equations. CMES Comput. Model. Eng. Sci. 89(3), 189–220 (2012)

    MathSciNet  MATH  Google Scholar 

  46. Tian, Z.F., Ge, Y.B.: A fourth-order compact ADI method for solving two-dimensional unsteady convection–diffusion problems. J. Comput. Appl. Math. 198(1), 268–286 (2007)

  47. Tien, C.M.T., Thai-Quang, N., Mai-Duy, N., Tran, C.-D., Tran-Cong, T.: A three-point coupled compact integrated RBF scheme for second-order differential problems. CMES Comput. Model. Eng. Sci. 104(6), 425–469 (2015)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  49. Walz, G.: Identities for trigonometric B-splines with an application to curve design. BIT Numer. Math. 37(1), 189–201 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  50. Yang, X.H., Zhang, H.X., Xu, D.: Orthogonal spline collocation method for the two-dimensional fractional sub-diffusion equation. J. Comput. Phys. 256, 824–837 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  51. Zaslavsky, G.M., Stevens, D., Weitzner, H.: Self-similar transport in incomplete chaos. Phys. Rev. E. 48(3), 1683–1694 (1993)

    Article  MathSciNet  Google Scholar 

  52. Zhu, X.G., Nie, Y.F., Wang, J.G., Yuan, Z.B.: A numerical approach for the Riesz space-fractional Fisher’ equation in two-dimensions. Int. J. Comput. Math. 94(2), 296–315 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  53. Zeng, F.H., Li, C.P., Liu, F.W., Turner, I.: The use of finite difference/element approaches for solving the time-fractional subdiffusion equation. SIAM J. Sci. Comput. 35(6), A2976–A3000 (2013)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  55. Zhai, S.Y., Feng, X.L.: A block-centered finite-difference method for the time-fractional diffusion equation on nonuniform grids. Numer. Heat Transf. Part B Fundam. 69(3), 217–233 (2016)

    Article  Google Scholar 

  56. Zhou, F.Y., Xu, X.Y.: The third kind Chebyshev wavelets collocation method for solving the time-fractional convection diffusion equations with variable coefficients. Appl. Math. Comput. 280, 11–29 (2016)

    MathSciNet  Google Scholar 

  57. Zhuang, P.H., Liu, F.W.: Implicit difference approximation for the time fractional diffusion equation. J. Appl. Math. Comput. 22(3), 87–99 (2006)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors are very grateful to the reviewers for their valuable comments and suggestions. This research was supported by National Natural Science Foundations of China (Nos. 11471262 and 11501450).

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  1. Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an, 710129, Shaanxi, People’s Republic of China

    X. G. Zhu, Y. F. Nie & W. W. Zhang

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Correspondence to Y. F. Nie.

Appendix: The explicit formulas of the fractional derivatives of cubic B-splines

Appendix: The explicit formulas of the fractional derivatives of cubic B-splines

The fractional derivatives center at \(x_{-1}\), \(x_0\), and \(x_1\):

$$\begin{aligned} {^{\mathrm{RL}}_{x_0}}D^\beta _xB_{-1}(x)&=\left\{ \begin{array}{l} \frac{(1-\beta )(x-x_0)^{-\beta }}{\Gamma (2-\beta )}-\frac{3(x-x_0)^{1-\beta }}{\Gamma (2-\beta )h} +\frac{6(x-x_0)^{2-\beta }}{\Gamma (3-\beta )h^2} -\frac{6(x-x_0)^{3-\beta }}{\Gamma (4-\beta )h^3}, \quad x \in [x_0,x_1)\\ \frac{(1-\beta )(x-x_0)^{-\beta }}{\Gamma (2-\beta )}-\frac{3(x-x_0)^{1-\beta }}{\Gamma (2-\beta )h} +\frac{6(x-x_0)^{2-\beta }}{\Gamma (3-\beta )h^2} -\frac{6(x-x_0)^{3-\beta }}{\Gamma (4-\beta )h^3} +\frac{6(x-x_1)^{3-\beta }}{\Gamma (4-\beta )h^3}, \quad x \in [x_1,x_M] \end{array} \right. \\ {^{\mathrm{RL}}_{x_0}}D^\beta _xB_0(x)&=\left\{ \begin{array}{l} \frac{4(1-\beta )(x-x_0)^{-\beta }}{\Gamma (2-\beta )}-\frac{12(x-x_0)^{2-\beta }}{\Gamma (3-\beta )h^2} +\frac{18(x-x_0)^{3-\beta }}{\Gamma (4-\beta )h^3}, \quad x \in [x_0,x_1)\\ \frac{4(1-\beta )(x-x_0)^{-\beta }}{\Gamma (2-\beta )}-\frac{12(x-x_0)^{2-\beta }}{\Gamma (3-\beta )h^2} +\frac{18(x-x_0)^{3-\beta }}{\Gamma (4-\beta )h^3}-\frac{24(x-x_1)^{3-\beta }}{\Gamma (4-\beta )h^3}, \quad x \in [x_1,x_2)\\ \frac{4(1-\beta )(x-x_0)^{-\beta }}{\Gamma (2-\beta )}-\frac{12(x-x_0)^{2-\beta }}{\Gamma (3-\beta )h^2} +\frac{18(x-x_0)^{3-\beta }}{\Gamma (4-\beta )h^3}-\frac{24(x-x_1)^{3-\beta }}{\Gamma (4-\beta )h^3} +\frac{6(x-x_2)^{3-\beta }}{\Gamma (4-\beta )h^3}, \quad x \in [x_2,x_M] \end{array} \right. \\ {^{\mathrm{RL}}_{x_0}}D^\beta _xB_1(x)&=\left\{ \begin{array}{l} \frac{(1-\beta )(x-x_0)^{-\beta }}{\Gamma (2-\beta )}+\frac{3(x-x_0)^{1-\beta }}{\Gamma (2-\beta )h} +\frac{6(x-x_0)^{2-\beta }}{\Gamma (3-\beta )h^2} -\frac{18(x-x_0)^{3-\beta }}{\Gamma (4-\beta )h^3}, \quad x \in [x_0,x_1)\\ \frac{(1-\beta )(x-x_0)^{-\beta }}{\Gamma (2-\beta )}+\frac{3(x-x_0)^{1-\beta }}{\Gamma (2-\beta )h} +\frac{6(x-x_0)^{2-\beta }}{\Gamma (3-\beta )h^2} -\frac{18(x-x_0)^{3-\beta }}{\Gamma (4-\beta )h^3} +\frac{36(x-x_1)^{3-\beta }}{\Gamma (4-\beta )h^3}, \quad x \in [x_1,x_2)\\ \frac{(1-\beta )(x-x_0)^{-\beta }}{\Gamma (2-\beta )}+\frac{3(x-x_0)^{1-\beta }}{\Gamma (2-\beta )h} +\frac{6(x-x_0)^{2-\beta }}{\Gamma (3-\beta )h^2} -\frac{18(x-x_0)^{3-\beta }}{\Gamma (4-\beta )h^3} +\frac{36(x-x_1)^{3-\beta }}{\Gamma (4-\beta )h^3} -\frac{24(x-x_2)^{3-\beta }}{\Gamma (4-\beta )h^3},\quad x \in [x_2,x_3)\\ \frac{(1-\beta )(x-x_0)^{-\beta }}{\Gamma (2-\beta )}+\frac{3(x-x_0)^{1-\beta }}{\Gamma (2-\beta )h} +\frac{6(x-x_0)^{2-\beta }}{\Gamma (3-\beta )h^2} -\frac{18(x-x_0)^{3-\beta }}{\Gamma (4-\beta )h^3} +\frac{36(x-x_1)^{3-\beta }}{\Gamma (4-\beta )h^3}-\frac{24(x-x_2)^{3-\beta }}{\Gamma (4-\beta )h^3} +\frac{6(x-x_3)^{3-\beta }}{\Gamma (4-\beta )h^3}.\quad x \in [x_3,x_M] \end{array} \right. \end{aligned}$$

The fractional derivatives center at \(x_{m}\) with \(2\le m\le M+1\):

$$\begin{aligned} {^{\mathrm{RL}}_{x_0}}D^\beta _xB_m(x)=\left\{ \begin{array}{l} 0,\qquad \qquad \qquad \, x \in [x_0,x_{m-2})\\ \frac{6(x-x_{m-2})^{3-\beta }}{\Gamma (4-\beta )h^3}, \quad x \in [{x_{m - 2}},{x_{m - 1}})\\ \frac{6(x-x_{m-2})^{3-\beta }}{\Gamma (4-\beta )h^3}-\frac{24(x-x_{m-1})^{3-\beta }}{\Gamma (4-\beta )h^3}, \quad x \in [{x_{m - 1}},{x_m})\\ \frac{6(x-x_{m-2})^{3-\beta }}{\Gamma (4-\beta )h^3}-\frac{24(x-x_{m-1})^{3-\beta }}{\Gamma (4-\beta )h^3} +\frac{36(x-x_{m})^{3-\beta }}{\Gamma (4-\beta )h^3}, \quad x \in [{x_m},{x_{m + 1}})\\ \frac{6(x-x_{m-2})^{3-\beta }}{\Gamma (4-\beta )h^3}-\frac{24(x-x_{m-1})^{3-\beta }}{\Gamma (4-\beta )h^3} +\frac{36(x-x_{m})^{3-\beta }}{\Gamma (4-\beta )h^3}-\frac{24(x-x_{m+1})^{3-\beta }}{\Gamma (4-\beta )h^3}, \quad x \in [{x_{m + 1}},{x_{m + 2}})\\ \frac{6(x-x_{m-2})^{3-\beta }}{\Gamma (4-\beta )h^3}-\frac{24(x-x_{m-1})^{3-\beta }}{\Gamma (4-\beta )h^3} +\frac{36(x-x_{m})^{3-\beta }}{\Gamma (4-\beta )h^3}-\frac{24(x-x_{m+1})^{3-\beta }}{\Gamma (4-\beta )h^3} +\frac{6(x-x_{m+2})^{3-\beta }}{\Gamma (4-\beta )h^3}, \quad x \in [x_{m+2},x_M] \end{array} \right. \end{aligned}$$

which contain \({^{\mathrm{RL}}_{x_0}}D^\beta _xB_{M-1}(x)\), \({^{\mathrm{RL}}_{x_0}}D^\beta _xB_M(x)\), and \({^{\mathrm{RL}}_{x_0}}D^\beta _xB_{M+1}(x)\) as special cases:

$$\begin{aligned} {^{\mathrm{RL}}_{x_0}}D^\beta _xB_{M-1}(x)&=\left\{ \begin{array}{l} 0,\qquad \qquad \qquad \, x \in [x_0,x_{M-3})\\ \frac{6(x-x_{M-3})^{3-\beta }}{\Gamma (4-\beta )h^3}, \quad x \in [x_{M - 3},x_{M - 2})\\ \frac{6(x-x_{M-3})^{3-\beta }}{\Gamma (4-\beta )h^3}-\frac{24(x-x_{M-2})^{3-\beta }}{\Gamma (4-\beta )h^3}, \quad x \in [{x_{M - 2}},x_{M-1})\\ \frac{6(x-x_{M-3})^{3-\beta }}{\Gamma (4-\beta )h^3}-\frac{24(x-x_{M-2})^{3-\beta }}{\Gamma (4-\beta )h^3} +\frac{36(x-x_{M-1})^{3-\beta }}{\Gamma (4-\beta )h^3}, \quad x \in [x_{M-1},x_M] \end{array} \right. \\ {^{\mathrm{RL}}_{x_0}}D^\beta _xB_M(x)&=\left\{ \begin{array}{l} 0, \quad x \in [x_0,x_{M-2})\\ \frac{6(x-x_{M-2})^{3-\beta }}{\Gamma (4-\beta )h^3}, \quad x \in [x_{M-2},x_{M-1})\\ \frac{6(x-x_{M-2})^{3-\beta }}{\Gamma (4-\beta )h^3}-\frac{24(x-x_{M-1})^{3-\beta }}{\Gamma (4-\beta )h^3}, \quad x \in [x_{M-1},x_{M}] \end{array} \right. \\ {^{\mathrm{RL}}_{x_0}}D^\beta _xB_{M+1}(x)&=\left\{ \begin{array}{l} 0, \quad x \in [x_0,x_{M-1})\\ \frac{6(x-x_{M-1})^{3-\beta }}{\Gamma (4-\beta )h^3}. \quad x \in [x_{M-1},x_M] \end{array} \right. \end{aligned}$$

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Zhu, X.G., Nie, Y.F. & Zhang, W. . An efficient differential quadrature method for fractional advection–diffusion equation. Nonlinear Dyn 90, 1807–1827 (2017). https://doi.org/10.1007/s11071-017-3765-x

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