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Journal of Scientific Computing

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14-03-2018

The Unstructured Mesh Finite Element Method for the Two-Dimensional Multi-term Time–Space Fractional Diffusion-Wave Equation on an Irregular Convex Domain

Authors: Wenping Fan, Xiaoyun Jiang, Fawang Liu, Vo Anh

Published in: Journal of Scientific Computing | Issue 1/2018

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Abstract

In this paper, the two-dimensional multi-term time-space fractional diffusion-wave equation on an irregular convex domain is considered as a much more general case for wider applications in fluid mechanics. A novel unstructured mesh finite element method is proposed for the considered equation. In most existing works, the finite element method is applied on regular domains using uniform meshes. The case of irregular convex domains, which would require subdivision using unstructured meshes, is mostly still open. Furthermore, the orders of the multi-term time-fractional derivatives have been considered to belong to (0, 1] or (1, 2] separately in existing models. In this paper, we consider two-dimensional multi-term time-space fractional diffusion-wave equations with the time fractional orders belonging to the whole interval (0, 2) on an irregular convex domain. We propose to use a mixed difference scheme in time and an unstructured mesh finite element method in space. Detailed implementation and the stability and convergence analyses of the proposed numerical scheme are given. Numerical examples are conducted to evaluate the theoretical analysis.

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Al-Refai, M., Luchko, Y.: Maximum principle for the multi-term timefractional diffusion equations with the Riemann–Liouville fractional derivatives. Appl. Math. Comput. 257, 40–51 (2015)MathSciNetMATH

Bu, W., Liu, X., Tang, Y., Yang, J.: Finite element multigrid method for multiterm time fractional advection diffusion equations. Int. J. Model. Simul. Sci. Comput. 6(1), 1540001 (2015)CrossRef

Bu, W., Tang, Y., Wu, Y., Yang, J.: Finite difference/finite element method for two-dimensional space and time fractional Bloch–Torrey equations. J. Comput. Phys. 293, 264–279 (2015)MathSciNetCrossRefMATH

Bu, W., Tang, Y., Yang, J.: Galerkin finite element method for two-dimensional Riesz space fractional diffusion equations. J. Comput. Phys. 276, 26–38 (2014)MathSciNetCrossRefMATH

Cowper, G.R.: Gaussian quadrature formulas for triangles. Int. J. Numer. Methods Eng. 7(3), 405–408 (1973)CrossRefMATH

Cristescu, M., Loubignac, G.: Gaussian Quadrature Formulas for Functions with Singularities in 1/R Over Triangles and Quadrangles. Pentech Press, London (1978)MATH

Ervin, V.J., Roop, J.P.: Variational formulation for the stationary fractional advection dispersion equation. Numer. Methods Partial Differ. Equ. 22(3), 558–576 (2006)MathSciNetCrossRefMATH

Ervin, V.J., Roop, J.P.: Variational solution of fractional advection dispersion equations on bounded domains in \({\mathbb{R}}^{d}\). Numer. Method Partial Differ. Equ. 23, 256–281 (2007)CrossRefMATH

Fan, W., Jiang, X., Chen, S.: Parameter estimation for the fractional fractal diffusion model based on its numerical solution. Comput. Math. Appl. 71(2), 642–651 (2016)MathSciNetCrossRef

10.

Fan, W., Liu, F., Jiang, X., Turner, I.: A novel unstructured mesh finite element method for solving the time-space fractional wave equation on a two-dimensional irregular convex domain. Fract. Calc. Appl. Anal. 20(2), 352–383 (2017)MathSciNetCrossRefMATH

11.

Feng, L., Liu, F., Ian, T.: Novel numerical analysis of multi-term time fractional viscoelastic non-Newtonian fluid models for simulating unsteady MHD Couette flow of a generalized Oldroyd-B fluid. arXiv preprint arXiv:1710.09976 (2017)

12.

Fetecau, C., Athar, M., Fetecau, C.: Unsteady flow of a generalized Maxwell fluid with fractional derivative due to a constantly accelerating plate. Comput. Math. Appl. 57, 596–603 (2009)MathSciNetCrossRefMATH

13.

Gao, G.h., Alikhanov, A.A., Sun, Z.z.: The temporal second order difference schemes based on the interpolation approximation for solving the time multi-term and distributed-order fractional sub-diffusion equations. J. Sci. Comput. 73(1), 93–121 (2017)

14.

Gatto, P., Hesthaven, J.S.: Numerical approximation of the fractional Laplacian via hp-finite elements, with an application to image denoising. J. Sci. Comput. 65(1), 249–270 (2015)MathSciNetCrossRefMATH

15.

Geuzaine, C., Remacle, J.F.: Gmsh: a 3-D finite element mesh generator with built-in pre-and post-processing facilities. Int. J. Numer. Methods Eng. 79(11), 1309–1331 (2009)MathSciNetCrossRefMATH

16.

Hilfer, R.: Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000)CrossRefMATH

17.

Jiang, H., Liu, F., Turner, I., Burrage, K.: Analytical solutions for the multi-term time-space Caputo–Riesz fractional advection-diffusion equations on a finite domain. J. Math. Anal. Appl. 389(2), 1117–1127 (2012)MathSciNetCrossRefMATH

18.

Jiang, X., Qi, H.: Thermal wave model of bioheat transfer with modified Riemann–Liouville fractional derivative. J. Physics A Math. Theor 45(48), 485101 (2012)MathSciNetCrossRefMATH

19.

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)MathSciNetCrossRefMATH

20.

Khan, M., Maqbool, K., Hayat, T.: Influence of Hall current on the flows of a generalized Oldroyd-B fluid in a porous space. Acta Mech. 184(1), 1–13 (2006)CrossRefMATH

21.

Khana, M., Anjuma, A., Fetecau, C., Qi, H.: Exact solutions for some oscillating motions of a fractional Burgers’ fluid. Math. Comput. Model. 51, 682–692 (2010)MathSciNetCrossRef

22.

Li, Y., Pan, C., Meng, X., Ding, Y., Chen, H.: A method of approximate fractional order differentiation with noise immunity. Chemom. Intell. Lab. Syst. 144, 31–38 (2015)CrossRef

23.

Liu, F., Zhuang, P., Liu, Q.: Numerical Methods of Fractional Partial Differential Equations and Applications. Science Press, Beijing (2015). (in Chinese)

24.

Ming, C., Liu, F., Zheng, L., Turner, I., Anh, V.: Analytical solutions of multi-term time fractional differential equations and application to unsteady flows of generalized viscoelastic fluid. Comput. Math. Appl. 72(9), 2084–2097 (2016)MathSciNetCrossRefMATH

25.

Podlubny, I.: Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of their Solution and some of their Applications, vol. 198. Academic press, New York (1998)MATH

26.

Qi, H., Jin, H.: Unsteady rotating flows of a viscoelastic fluid with the fractional Maxwell model between coaxial cylinders. Acta Mech. Sin. 22, 301–305 (2006)CrossRefMATH

27.

Ren, J., Sun, Z.: Efficient and stable numerical methods for multi-term time fractional sub-diffusion equations. East Asian J. Appl. Math. 4(3), 242–266 (2014)MathSciNetCrossRefMATH

28.

Ren, J., Sun, Z.: Efficient numerical solution of the multi-term time fractional diffusion-wave equation. East Asian J. Appl. Math. 5(1), 1–28 (2015)MathSciNetCrossRefMATH

29.

Roop, J.P.: Variational solution of the fractional advection dispersion equation. Ph.D. thesis (2004)

30.

Vieru, D., Fetecau, C., Fetecau, C.: Flow of a viscoelastic fluid with the fractional Maxwell model between two side walls perpendicular to a plate. Appl. Math. Comput. 200, 459–464 (2008)MathSciNetMATH

31.

Xu, H., Jiang, X., Yu, B.: Numerical analysis of the space fractional Navier–Stokes equations. Appl. Math. Lett. 69, 94–100 (2017)MathSciNetCrossRefMATH

32.

Xue, C., Nie, J., Tan, W.: An exact solution of start-up flow for the fractional generalized Burgers fluid in a porous half-space. Nonlinear Anal. Theory Methods Appl. 69(7), 2086–2094 (2008)MathSciNetCrossRefMATH

33.

Yang, Z., Yuan, Z., Nie, Y., Wang, J., Zhu, X., Liu, F.: Finite element method for nonlinear Riesz space fractional diffusion equations on irregular domains. J. Comput. Phys. 330, 863–883 (2017)MathSciNetCrossRefMATH

34.

Zhang, H., Liu, F., Anh, V.: Galerkin finite element approximation of symmetric space-fractional partial differential equations. Appl. Math. Comput. 217(6), 2534–2545 (2010)MathSciNetMATH

35.

Zhao, Y., Chen, P., Bu, W., Liu, X., Tang, Y.: Two mixed finite element methods for time-fractional diffusion equations. J. Sci. Comput. 70(1), 407–428 (2017)MathSciNetCrossRefMATH

36.

Zhao, Z., Zheng, Y., Guo, P.: A galerkin finite element method for a class of time-space fractional differential equation with nonsmooth data. J. Sci. Comput. 70(1), 386–406 (2017)MathSciNetCrossRefMATH

37.

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

38.

Zhuang, P., Liu, F., Turner, I., Gu, Y.: Finite volume and finite element methods for solving a one-dimensional space-fractional Boussinesq equation. Applied Mathematical Modelling 38(15–16), 3860–3870 (2014)MathSciNetCrossRef

Title: The Unstructured Mesh Finite Element Method for the Two-Dimensional Multi-term Time–Space Fractional Diffusion-Wave Equation on an Irregular Convex Domain
Authors: Wenping Fan
Xiaoyun Jiang
Fawang Liu
Vo Anh
Publication date: 14-03-2018
Publisher: Springer US
Published in: Journal of Scientific Computing / Issue 1/2018
Print ISSN: 0885-7474
Electronic ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-018-0694-x

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