Abstract
A novel multi-term time-fractional mixed diffusion and diffusion-wave equation will be considered in this work. Different from the general multi-term time-fractional mixed diffusion and diffusion-wave equations, this new multi-term equation possesses a special time-fractional operator on the spatial derivative. We use a new discrete scheme to approximate the time-fractional derivative, which can improve the temporal accuracy. Then, a fully discrete spectral scheme is developed based on finite difference discretization in time and Legendre spectral approximation in space. Meanwhile, a very important lemma is proposed and proved, to obtain the unconditional stability and convergence of the fully discrete spectral scheme. Finally, four numerical experiments are presented to confirm our theoretical analysis. Both of our analysis and numerical test indicate that the fully discrete scheme is accurate and efficient in solving the generalized multi-term time-fractional mixed diffusion and diffusion-wave equation.
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References
Metzler, R., Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 39, 1–77 (2000)
Sun, H.G., Zhang, Y., Baleanu, D., Chen, W., Chen, Y.Q.: A new collection of real world applications of fractional calculus in science and engineering. Commun. Nonlinear Sci. Numer. Simul. 64, 213–231 (2018)
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies. Elsevier, Amsterdam (2006)
Diethelm, K.: The Analysis of Fractional Differential Equations. Springer, Berlin (2010)
Du, Q., Gunzburger, M., Lehoucq, R.B., Zhou, K.: Analysis and approximation of nonlocal diffusion problems with volume constrains. SIAM Rev. 54, 667–696 (2012)
Baleanu, D., Jajarmi, A., Bonyah, E., Hajipour, M.: New aspects of the poor nutrition in the life cycle within the fractional calculus. Adv. Differ. Equ. 2018, 230 (2018)
Arafa, A.A.M., Rida, S.Z., Mohammadein, A.A., Ali, H.M.: Solving nonlinear fractional differential equation by generalized Mittag-Leffler function method. Commun. Theor. Phys. 59, 661–663 (2013)
Abbaszadeh, M.: Error estimate of second-order finite difference scheme for solving the Riesz space distributed-order diffusion equation. Appl. Math. Lett. 88, 179–185 (2019)
Arshad, S., Bu, W.P., Huang, J.F., Tang, Y.F., Zhao, Y.: Finite difference method for time space linear and nonlinear fractional diffusion equations. Int. J. Comput. Math. 95, 202–217 (2018)
Zeng, F., Turner, I., Burrage, K.: A stable fast time-stepping method for fractional integral and derivative operator. J. Sci. Comput. 77, 283–307 (2018)
Celik, C., Duman, M.: Finite element method for a symmetric tempered fractional diffusion equation. Appl. Numer. Math. 120, 270–286 (2017)
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, 1270–1285 (2017)
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, 352–383 (2017)
Chakraborty, A., Rathish Kumar, B.V.: Finite element method for drifted space fractional tempered diffusion equation. J. Appl. Math. Comput. 61, 117–135 (2019)
Simmons, A., Yang, Q.Q., Moroney, T.: A finite volume method for two-sided fractional diffusion equations on non-uniform meshes. J. Comput. Phys. 335, 747–759 (2017)
Li, J., Liu, F., Feng, L.B., Turner, I.: A novel finite volume method for the Riesz space distributed-order advection–diffusion equation. Appl. Math. Model. 46, 536–553 (2017)
Zheng, M., Liu, F., Turner, I., Anh, V.: A novel high order space-time spectral method for the time frctional Fokker–Planck equation. SIAM J. Sci. Comput. 37, A701–A724 (2015)
Zeng, F., Liu, F., Li, C., 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, 2599–2622 (2014)
Aayernouri, M., Karniadakis, G.E.: Discontinuous spectral element methods for time- and space-fractional advection equations. SIAM J. Sci. 36, B684–B707 (2014)
Bhrawy, A.H., Baleanu, D.: A spectral Legendre–Gauss–Lobatto collocation method for a space-fractional advection diffusion equations with variable coefficients. Rep. Math. Phys. 72, 219–233 (2013)
Dehghan, M., Abbaszadeh, M.: A legendre spectral element method (SEM) based on the modified bases for solving neutral delay distributed-order fractional damped diffusion-wave equation. Math. Methods Appl. Sci. 41, 3476–3494 (2018)
Abbaszadeh, M., Dehghan, M.: An improved meshless method for solving two-dimensional distributed order time-fractional diffusion-wave equation with error estimate. Numer. Algorithms 75, 172–211 (2017)
Agheli, B.: Solving fractional partial differential equation by using wavelet operational method. J. Math. Comput. Sci. 7, 234–240 (2013)
Hyder Ali, S., Shah, Muttaqi: Some accelerated flows of a generalized Oldroyd-B fluid between two side walls perpendicular to the plate. Nonlinear Anal. Real World Appl. 10, 2146–2150 (2009)
Qi, H.T., Xu, M.Y.: Some unsteady unidirectional flows of a generalized Oldroyd-B fluid with fractional derivative. Appl. Math. Model. 33, 4184–4191 (2009)
Liu, Y.Q., Zheng, L.C., Zhang, X.X.: Unsteady MHD Couette flow of a generalized Oldroyd-B fluid with fractional derivative. Comput. Math. Appl. 61, 443–450 (2011)
Zheng, L.C., Liu, Y.Q., Zhang, X.X.: Slip effects on MHD flow of a generalized Oldroyd-B fluid with fractional derivative. Nonlinear Anal. Real World Appl. 13, 513–523 (2012)
Zhao, J., Zheng, L., Zhang, X.X., Liu, F., Chen, X.: Unsteady natural convection heat transfer past a vertical flat plate embedded in a porous medium saturated with fractional Oldroyd-B fluid. J. Heat Transf. 139, 012501 (2017)
Jiang, Y., Qi, H., Xu, H., Jiang, X.: Transient electroosmotic slip flow of fractional Oldroyd-B fluids. Microfluid Nanofluid 21, 1–10 (2017)
Hayat, T., Zaid, S., Asghar, S., Hendi, A.A.: Exact solutions in generalized Oldroyd-B fluid. Appl. Math. Mech. Engl. Edn. 33, 411–426 (2012)
Khan, M., Hayat, T., Asghar, S.: Exact solution for MHD flow of a generalized Oldroyd-B fluid with modified Darcy’s law. Int. J. Eng. Sci. 44, 333–339 (2006)
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, 1–13 (2006)
Abro, K.A., Hussain, M., Baig, M.M.: Slippage of magnetohydrodynamic fractional generalized Oldroyd-B fluid in porous medium. Prog. Fract. Differ. Appl. 3, 69–80 (2017)
Sun, Z.Z., Wu, X.N.: A fully discrete difference scheme for a diffusion-wave system. Appl. Numer. Math. 56, 193–209 (2006)
Henry, B.I., Langlands, T.A.M., Wearne, S.L.: Fractional cable models for spiny neuronal dendrites. Phys. Rev. Lett. 100, 128103 (2008)
Momani, S.: Analytic and approximate solutions of the space- and time-fractional telegraph equations. Appl. Math. Comput. 170, 1126–1134 (2005)
Ming, C., Liu, F., Zheng, L., Turner, I., Anh, V.: Analytical solutions of multi-term time fracdtional differential equations and application to unsteady flows of generalized viscoelastic fluid. Comput. Math. Appl. 72, 2084–2097 (2016)
Jiang, X.Y., Qi, H.T.: Thermal wave model of bioheat transfer with modified Riemann–Lioubille fractional derivative. J. Phys. A Math. Theor. 45, 831–842 (2012)
Bazhlekova, E., Bazhlekov, I.: Viscoelastic flows with fractional derivative models: computational approach by convolutional calculus of Dimovski. Fract. Calc. Appl. Anal. 17, 954–976 (2014)
Feng, L.B., Liu, F., Turner, I., Zhuang, P.: Numerical methods and analysis for simulating the flow of a generalized Oldroyd-B fluid between two infinite parallel rigid plates. Int. J. Heat Mass Transf. 115, 1309–1320 (2017)
Feng, L.B., Liu, F., Turner, I.: Novel numerical analysis of multi-term time frctional viscoelastic non-Newtonian fluid models for simulating unsteady MHD couette flow of a generalized Oldroyd-B fluid. Fract. Calc. Appl. Anal. 21, 1073–1103 (2018)
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)
Liu, Z.T., Lü, S.J., Liu, F.W.: Fully discrete spectral methods for solving time fractional nonlinear Sine-Gordon equation with smooth and non-smooth solutions. Appl. Math. Comput. 333, 213–224 (2018)
Bernardi, C., Maday, Y.: Approximations Spectrales de problems aux Limites Elliptiques. Spring, Berlin (1992)
Alikhanov, A.A.: A new difference scheme for the time fractional diffusion equation. J. Comput. Phys. 280, 424–438 (2015)
Liu, Y., Liu, F., Feng, L., Xin, B.: Novel numerical analysis for simulating the generalized 2D multi-term time fractional Oldroyd-B fluid model. arXiv:1903.07816
Lopez-Marcos, J.C.: A difference scheme for a nonlinear partial integrodifferential equation. SIAM J. Numer. Anal. 27, 20–31 (1990)
Böttcher, A., Silbermann, B.: Analysis of Toeplitz Operators, 2nd edn. Springer, Berlin (2005)
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This work is supported by National Natural Science Foundation of China (Nos. 11801060, 11572112 and 11972148).
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Liu, Y., Sun, H., Yin, X. et al. Fully discrete spectral method for solving a novel multi-term time-fractional mixed diffusion and diffusion-wave equation. Z. Angew. Math. Phys. 71, 21 (2020). https://doi.org/10.1007/s00033-019-1244-6
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DOI: https://doi.org/10.1007/s00033-019-1244-6
Keywords
- Multi-term time-fractional derivative
- Mixed diffusion equations
- Legendre spectral method
- Finite difference discretization
- Stability and convergence