Efficient numerical schemes for the solution of generalized time fractional Burgers type equations. (English) Zbl 1394.65106
The authors propose two semi-implicit Fourier pseudospectral schemes for the solution of generalized time-fractional Burgers-type equations. The consistency, stability, and convergence analysis of the schemes are analyzed and an unconditional convergence is proved. Some numerical results are presented using the fast Fourier transform (FFT) and a comparison between these results and some recent reported results is provided. These numerical experiments illustrate that the order of accuracy of the proposed schemes are in good agreement with the obtained theoretical ones.
Reviewer: Abdallah Bradji (Annaba)
MSC:
| 65M70 | Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs |
| 35Q53 | KdV equations (Korteweg-de Vries equations) |
| 35R11 | Fractional partial differential equations |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |
| 65T50 | Numerical methods for discrete and fast Fourier transforms |
| 35R60 | PDEs with randomness, stochastic partial differential equations |
Keywords:
stochastic time-fractional Burgers equation; Fourier pseudospectral; consistency; stability; convergence; additive noiseReferences:
| [1] | Abdel-Salam, E.A.B., Hassan, G.F.: Multi-wave solutions of the space-time fractional Burgers and Sharma-Tasso-Olver equations. Ain Shams Eng. J. 7, 463-472 (2016) · doi:10.1016/j.asej.2015.04.001 |
| [2] | Abdel-Salam, E.A.B., Yousif, E.A., Arko, Y.A.S., Gumma, E.A.E.: Solution of moving boundary space-time fractional Burger’s equation. J. Appl. Math. 2014, Article ID 218092, 8 pages (2014) · Zbl 1437.35604 · doi:10.1155/2014/218092 |
| [3] | Bhrawy, A.H., Zaky, M.A., Baleanu, D.: New numerical approximations for space-time fractional Burgers’ equations via a Legendre spectral-collocation method. Roman. Rep. Phys. 67, 340-349 (2015) |
| [4] | Cameron, R.H., Martin, W.T.: The orthogonal development of nonlinear functionals in series of Fourier-Hermite functionals. Ann. Math. 48, 385-392 (1947) · Zbl 0029.14302 · doi:10.2307/1969178 |
| [5] | Canuto, C., Quarteroni, A.: Approximation results for orthogonal polynomials in Sobolev spaces. Math. Comput. 38, 67-86 (1982) · Zbl 0567.41008 · doi:10.1090/S0025-5718-1982-0637287-3 |
| [6] | Cheng, K., Feng, W., Gottlieb, S., Wang, C.: A Fourier pseudospectral method for the “Good” Boussinesq equation with second-order temporal accuracy. Numer. Methods Partial Differ. Eq. 31, 202-224 (2015) · Zbl 1327.65195 · doi:10.1002/num.21899 |
| [7] | Dehghan, M., Manafian, J., Saadatmandi, A.: Solving nonlinear fractional partial differential equations using the homotopy analysis method. Numer. Methods Partial Differ. Eq. 26, 448-479 (2010) · Zbl 1185.65187 |
| [8] | Diethelm, K.: An algorithm for the numerical solution for differential equations of fractional order. Elec. Trans. Numer. Anal. 5, 1-6 (1997) · Zbl 0890.65071 |
| [9] | W.E: Convergence of Fourier methods for Navier-Stokes equations. SIAM J. Numer. Anal. 30, 650-674 (1993) · Zbl 0776.76021 · doi:10.1137/0730032 |
| [10] | El-Danaf, T.S., Hadhoud, A.R.: Parametric spline functions for the solution of the one time fractional Burgers’ equation. Appl. Math. Modell. 36, 4557-4564 (2012) · Zbl 1252.65175 · doi:10.1016/j.apm.2011.11.035 |
| [11] | Esen, A., Tasbozan, O.: Numerical solution of time fractional Burgers equation by cubic B-spline finite elements. Mediterr. J. Math. 13, 1325-1337 (2016) · Zbl 1351.65076 · doi:10.1007/s00009-015-0555-x |
| [12] | Esen, A., Tasbozan, O.: Numerical solution of time fractional Burgers equation. Acta Univ. Sapientiae, Mathematica 7, 167-185 (2015) · Zbl 1333.65108 |
| [13] | Esen, A., Bulut, F., Oru, O.: A unified approach for the numerical solution of time fractional Burgers type equations. Eur. Phys. J. Plus 131, 116 (2016) · Zbl 1354.65194 · doi:10.1140/epjp/i2016-16116-5 |
| [14] | Gottlieb, S., Wang, C.: Stability and convergence analysis of fully discrete Fourier collocation spectral method for 3-D viscous Burgers equation. J. Sci. Comput. 53, 102-128 (2012) · Zbl 1297.76126 · doi:10.1007/s10915-012-9621-8 |
| [15] | Guo, B.Y., Zou, J.: Fourier spectral projection method and nonlinear convergence analysis for Navier-Stokes equations. J. Math. Anal Appl. 282, 766-791 (2003) · Zbl 1040.35061 · doi:10.1016/S0022-247X(03)00254-3 |
| [16] | Guo-Cheng, W.: Lie Group Classifications and Non-differentiable Solutions for Time-Fractional Burgers Equation. Commun. Theor. Phys. 55, 1073-1076 (2011) · Zbl 1264.35178 · doi:10.1088/0253-6102/55/6/23 |
| [17] | Gupta, A.K., Saha Ray, S.: On the solutions of fractional Burgers-Fisher and generalized Fisher’s equations using two reliable methods. Int. J. Math. Math. Sci. 2014, Article ID 682910, 16 pages (2014) · Zbl 1310.65136 · doi:10.1155/2014/682910 |
| [18] | Inc, M.: The approximate and exact solutions of the space- and time-fractional burgers equation with initial conditions by variational iteration method. J. Math. Anal. Appl. 345, 476-484 (2008) · Zbl 1146.35304 · doi:10.1016/j.jmaa.2008.04.007 |
| [19] | Jin, B., Lazarov, R., Zhou, Z.: An analysis of the L1 scheme for the subdiffusion equation with nonsmooth data. IMA J. Numer. Anal. 36, 197-221 (2016) · Zbl 1336.65150 |
| [20] | Jin, B., Zhou, Z.: An analysis of the Galerkin proper orthogonal decomposition for subdiffusion, ESAIM: Math. Modeling Numer. Anal. (in press). arXiv:1508.06134tic · Zbl 1365.65224 |
| [21] | Kalpinelli, E.A., Frangos, N.E., Yannacopoulos, A.N.: Numerical methods for hyperbolic SPDEs: a Wiener chaos approach. Stoch PDE Anal. Comp. 4, 606-633 (2013) · Zbl 1332.60105 · doi:10.1007/s40072-013-0019-x |
| [22] | Kang, X., Cheng, K., Guo, C.: A second-order Fourier pseudospectral method for the generalized regularized long wave equation. Adv. Diff. Eq. 2015, 339 (2015) · Zbl 1422.65282 · doi:10.1186/s13662-015-0676-3 |
| [23] | Khan, N.A., Ara, A., Mahmood, A.: Numerical solutions of time-fractional Burger equations: a comparison between generalized transformation technique with homotopy perturbation method. Int. J. Num. Method Heat Fluid Flow 22, 175-93 (2012) · Zbl 1357.65208 · doi:10.1108/09615531211199818 |
| [24] | Lakestani, M., Dehghan, M.: Numerical solutions of the generalized Kuramoto-Sivashinsky equation using B-spline functions. Appl. Math. Modell. 36, 605-617 (2012) · Zbl 1236.65114 · doi:10.1016/j.apm.2011.07.028 |
| [25] | Li, D., Zhang, C., Ran, M.: A linear finite difference scheme for generalized time fractional Burgers equation. Appl. Math. Modell. 40, 6069-6081 (2016) · Zbl 1465.65075 · doi:10.1016/j.apm.2016.01.043 |
| [26] | Lin, Y., Xu, C.: Finite difference/spectral approximations for the time-fractional diffusion equation. J. Comput. Phys. 225, 1533-1552 (2007) · Zbl 1126.65121 · doi:10.1016/j.jcp.2007.02.001 |
| [27] | Lototsky, S.; Rozovsky, B.; Kabanov, Y. (ed.); Liptser, R. (ed.); Stoyanov, J. (ed.), Stochastic differential equations: a Wiener chaos approach, 433-507 (2006), Berlin · Zbl 1178.60048 · doi:10.1007/978-3-540-30788-4_23 |
| [28] | Lue, W.: Wiener chaos expansion and numerical solutions of stochastic partial differential equations. PhD Thesis, California Institute of Technology, Pasadena (2006) · Zbl 0776.76021 |
| [29] | Mohebbi, A., Abbaszadeh, M.: Compact finite difference scheme for the solution of time fractional advection-dispersion equation. Numer. Algor. 63, 431-452 (2013) · Zbl 1380.65170 · doi:10.1007/s11075-012-9631-5 |
| [30] | Quarteroni, A., Valli, A.: Numerical approximation of partial differential equations. Springer Series in Computational Mathematics, vol. 23. Springer-Verlag (1994) · Zbl 0803.65088 |
| [31] | Tadmor, E.: Convergence of spectral methods to nonlinear conservation laws. SIAM J. Numer. Anal. 26, 30-44 (1989) · Zbl 0667.65079 · doi:10.1137/0726003 |
| [32] | Tadmor, E.: The exponential accuracy of Fourier and Chebyshev differencing methods. SIAM J. Numer. Anal. 23, 1-10 (1986) · Zbl 0613.65017 · doi:10.1137/0723001 |
| [33] | Sahoo, S., SahaRay, S.: New approach to find exact solutions of time-fractional Kuramoto-Sivashinsky equation. Physica A 434, 240-245 (2015) · Zbl 1400.35225 · doi:10.1016/j.physa.2015.04.018 |
| [34] | Song, L., Zhang, H.Q.: Application of homotopy analysis method to fractional KDV-Burgers-Kuramoto equation. Phys. Lett. A. 367, 88-94 (2007) · Zbl 1209.65115 · doi:10.1016/j.physleta.2007.02.083 |
| [35] | Sugimoto, N.: Burgers equation with a fractional derivative; hereditary effects on nonlinear acoustic waves. J. Fluild Mech. 225, 631-653 (1991) · Zbl 0721.76011 · doi:10.1017/S0022112091002203 |
| [36] | Sugimoto, N.: Generalized Burgers equation and fractional calculus. In: Nonlinear wave motion. Longman Scientic and Technical (1989) · Zbl 0674.35079 |
| [37] | Sun, Z.Z., Wu, X.: A fully discrete difference scheme for a diffusion-wave system. Appl. Numer. Math. 56, 193-209 (2006) · Zbl 1094.65083 · doi:10.1016/j.apnum.2005.03.003 |
| [38] | Xiu, D., Karniadakis, G.E.: The Wiener-Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput. 24, 619-644 (2002) · Zbl 1014.65004 · doi:10.1137/S1064827501387826 |
| [39] | Wang, Q.: Numerical solutions for fractional kdv-burgers equation by adomian decomposition method. Appl. Math. Comput. 182(2), 1048-1055 (2006) · Zbl 1107.65124 |
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