Numerical solution of time fractional Burgers equation

[1] L. Debnath, Partial Differential Equations for Scientists and Engineers, Birkhäuser, Boston, 1997.10.1007/978-1-4899-2846-7Search in Google Scholar

[2] S. Kutluay, A. Esen, I. Dag, Numerical solutions of the Burgers’ equation by the least squares quadratic B-spline finite element method, J. Comp. Appl. Math., 167 (2004) 21-33.Search in Google Scholar

[3] L. R. T. Gardner, G. A. Gardner, A. Dogan, A Petrov-Galerkin finite element scheme for Burgers equation, Arab. J. Sci. Eng., 22 (1997) 99-109.Search in Google Scholar

[4] A. H. A. Ali, L. R. T. Gardner, G. A. Gardner, A Collocation method for Burgers equation using cubic splines, Comp. Meth. Appl. Mech. Eng., (1992) 325-337.10.1016/0045-7825(92)90088-2Search in Google Scholar

[5] K. R. Raslan, A collocation solution for Burgers equation using quadratic B-spline finite elements, Intern. J. Computer Math., 80 (2003) 931-938.Search in Google Scholar

[6] I. Dag, D. Irk, B. Saka, A numerical solution of Burgers equation using cubic B-splines, Appl. Math. Comput., 163 (2005) 199-211.10.1016/j.amc.2004.01.028Search in Google Scholar

[7] M. A. Ramadan, T. S. El-Danaf, F. E. I. Abd Alaal, Numerical solution of Burgers equation using septic B-splines, Chaos, Solitons and Fractals, 26 (2005) 795-804.10.1016/j.chaos.2005.01.054Search in Google Scholar

[8] N. Sugimoto, Burgers equation with a fractional derivative; hereditary effects on nonlinear acoustic waves, J. Fluid Mech., 225 (1991) 631-653.10.1017/S0022112091002203Search in Google Scholar

[9] P. Miskinis, Some Properties of Fractional Burgers Equation, Mathematical Modelling and Analysis, 7 (2002) 151-158.10.3846/13926292.2002.9637187Search in Google Scholar

[10] S. Momani, Non-perturbative analytical solutions of the space- and timefractional Burgers equations, Chaos, Solitons and Fractals, 28 (2006) 930-937.10.1016/j.chaos.2005.09.002Search in Google Scholar

[11] M. Inc, The approximate and exact solutions of the space- and timefractional Burgers equations with initial conditions by variational iteration method, J. Math. Anal. Appl. 345 (2008) 476-484.10.1016/j.jmaa.2008.04.007Search in Google Scholar

[12] C. Li, Y. Wang, Numerical algorithm based on Adomian decomposition for fractional differential equations, Computers and Mathematics with Applications, 57 (2009) 1672-1681.10.1016/j.camwa.2009.03.079Search in Google Scholar

[13] T. S. El-Danaf, A. R. Hadhoud, Parametric spline functions for the solution of the one time fractional Burgers equation, Applied Mathematical Modelling, 36 (2012) 4557-4564.10.1016/j.apm.2011.11.035Search in Google Scholar

[14] M. Younis, A. Zafar, Exact Solution to Nonlinear Differential Equations of Fractional Order via (G _ /G)-Expansion Method, Applied Mathematics, 5 (2014) 1-6.10.4236/am.2014.51001Search in Google Scholar

[15] D. L. Logan, A First Course in the Finite Element Method (Fourth Edition), Thomson, 2007.Search in Google Scholar

[16] K. B. Oldham, J. Spanier, The Fractional Calculus, Academic, New York, 1974.Search in Google Scholar

[17] A. Esen, Y. Ucar, N. Yagmurlu, O Tasbozan, A Galerkin Finite Element Method to Solve Fractional Diffusion and Fractional Diffusion-Wave Equations, Mathematical Modelling and Analysis, 18 (2013) 260-273.Search in Google Scholar

[18] O. Tasbozan, A. Esen, N. M. Yagmurlu, Y. Ucar, A Numerical Solution to Fractional Diffusion Equation for Force-Free Case, Abstr. Appl. Anal. (2013), http://dx.doi.org/10.1155/2013/187383, Article ID 187383, 6 pp.10.1155/2013/187383Search in Google Scholar

[19] F. Liu, P. Zhuang, I. Turner, K. Burrage, V. Anh, A new fractional finite volume method for solving the fractional diffusion equation, Applied Mathematical Modelling, 38 (2014) 3871-3878.Search in Google Scholar

[20] H. Zhang, F. Liu, P. Zhuang, I. Turner, V. Anh, Numerical analysis of a new space-time variable fractional order advection-dispersion equation, Applied Mathematics and Computation, 242 (2014) 541-550.Search in Google Scholar

[21] A. Pedas, E. Tamme, Numerical solution of nonlinear fractional differential equations by spline collocation method, Journal of Computational and Applied Mathematics, 255 (2014) 216-230.Search in Google Scholar

[22] S. Butera, M. D. Paola, Fractional differential equations solved by using Mellin transform, Commun. Nonlinear. Sci. Numer. Simulat., 19 (2014) 2220-2227.Search in Google Scholar

[23] J. Zhao, Jingyu Xiao, N. J. Ford, Collocation methods for fractional integro-differential equations with weakly singular kernels, Numer. Algor., 65 (2014) 723-743.Search in Google Scholar

[24] A. B. Orovio, D. Kay, K. Burrage, Fourier spectral methods for fractional in space reaction-diffusion equations, BIT Numer. Math., 54 (2014) 937-954.Search in Google Scholar

[25] S. Tatar, S. Ulusoye, An inverse source problem for a one-dimensional space-time fractional diffusion equation, Applicable Analysis, (2014).10.1080/00036811.2014.979808Search in Google Scholar

[26] F. F. Dou, Y. C. Hon, Numerical computation for backward timefractional diffusion equation, Engineering Analysis with Boundary Elements, 40 (2014) 138-146.10.1016/j.enganabound.2013.12.001Search in Google Scholar

[27] B. Cockburn, K. Mustapha, A hybridizable discontinuous Galerkin method for fractional diffusion problems, Numer. Math., (2014).10.1007/s00211-014-0661-xSearch in Google Scholar

[28] Y. Zhanga, Z. Sunb, H. Liaoc, Finite difference methods for the time fractional diffusion equation on non-uniform meshes, Journal of Computational Physics, 265 (2014) 195-210.10.1016/j.jcp.2014.02.008Search in Google Scholar

[29] A. Esen, Y. Ucar, M. Yagmurlu, O. Tasbozan, Solving Fractional Diffusion and Fractional Diffusion-Wave Equations by Petrov-Galerkin Finite Element Method, TWMS J. App. Eng. Math., 4 (2014) 155-168.10.1515/tmj-2015-0020Search in Google Scholar

[30] A. Esen, O. Tasbozan, An approach to Time Fractional Gas Dynamics Equation: Quadratic B-Spline Galerkin Method, Applied Mathematics and Computation, 261 (1) (2015) 330-336.10.1016/j.amc.2015.03.126Search in Google Scholar

[31] A. Esen, O. Tasbozan, Y. Ucar, N. M. Yagmurlu, A B-spline collocation method for solving fractional diffusion and fractional diffusion-wave equations, Tbilisi Mathematical Journal, 8 (2) (2015) 181-193.10.1515/tmj-2015-0020Search in Google Scholar

[32] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.Search in Google Scholar

[33] P. M. Prenter, Splines and Variasyonel Methods, New York, John Wiley, 1975. Search in Google Scholar