A numerical approach for a class of time-fractional reaction-diffusion equation through exponential B-spline method. (English) Zbl 1442.65213
Authors’ abstract: A numerical approach for a class of time-fractional reaction-diffusion equations through the exponential B-spline method is presented. The proposed scheme is a combination of the Crank-Nicolson method for the Caputo time derivative and the exponential B-spline method for the space derivative. The unconditional stability and convergence of the proposed scheme are presented. Several numerical examples are presented to illustrate the feasibility and efficiency of the proposed scheme.
Reviewer: Ahmed M. A. El-Sayed (Alexandria)
MSC:
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 35R11 | Fractional partial differential equations |
| 65M70 | Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs |
Keywords:
time-fractional reaction-diffusion equation; Caputo derivative; exponential B-spline method; stability; convergenceReferences:
| [1] | Baeumer, B.; Kovacs, M.; Meerschaert, Mm, Numerical solutions for fractional reaction-diffusion equations, Comput Math Appl, 55, 10, 2212-2226 (2008) · Zbl 1142.65422 · doi:10.1016/j.camwa.2007.11.012 |
| [2] | Baleanu, D., Fractional calculus: models and numerical methods (2012), Singapore: World Scientific, Singapore · Zbl 1248.26011 · doi:10.1142/8180 |
| [3] | Chandra, Srs; Kumar, M., Exponential B-spline collocation method for self-adjoint singularly perturbed boundary value problems, Appl Numer Math, 58, 10, 1572-1581 (2008) · Zbl 1157.65047 · doi:10.1016/j.apnum.2007.09.008 |
| [4] | Dag, I.; Ersoy, O., The exponential cubic B-spline algorithm for Fisher equation, Chaos Soliton Fract, 86, 101-106 (2016) · Zbl 1357.65150 · doi:10.1016/j.chaos.2016.02.031 |
| [5] | Ersoy, O.; Dag, I., Numerical solutions of the reaction diffusion system by using exponential cubic B-spline collocation algorithms, Open Phys, 13, 1, 414-427 (2015) · doi:10.1515/phys-2015-0047 |
| [6] | Gao, G.; Sun, Z., A compact finite difference scheme for the fractional sub-diffusion equations, J Comput Phys, 230, 3, 586-595 (2011) · Zbl 1211.65112 · doi:10.1016/j.jcp.2010.10.007 |
| [7] | Gong, C.; Bao, Wm; Tang, G.; Jiang, Yw; Liu, J., A domain decomposition method for time fractional reaction-diffusion equation, Sci World J (2014) · doi:10.1155/2014/681707 |
| [8] | Henry, Bi; Wearne, Sl, Fractional reaction-diffusion, Phys A, 276, 448-455 (2000) · doi:10.1016/S0378-4371(99)00469-0 |
| [9] | Hesameddini, E.; Asadollahifard, E., A new reliable algorithm based on the sinc function for the time fractional diffusion equation, Numer Algor, 72, 4, 893-913 (2016) · Zbl 1361.65081 · doi:10.1007/s11075-015-0073-8 |
| [10] | Hilfer, R., Applications of fractional calculus in physics (2000), New York: World Scientific Publishing, New York · Zbl 0998.26002 · doi:10.1142/3779 |
| [11] | Karatay, I.; Kale, N.; Bayramoglu, Sr, A new difference scheme for time fractional heat equations based on the Crank-Nicolson method, Frac Calc Appl Anal, 16, 4, 892-910 (2013) · Zbl 1312.65136 |
| [12] | Karatay, I.; Kale, N., A new difference scheme for time fractional cable equation and stability analysis, Int J Appl Math Res, 4, 1, 52-57 (2015) · doi:10.14419/ijamr.v4i1.3875 |
| [13] | Kilbas, Aa; Srivastva, Hm; Trujillo, Jj, Theory and applications of fractional differential equations (2006), Amsterdam: Elsevier, Amsterdam · Zbl 1092.45003 |
| [14] | Li, X., Operational method for solving fractional differential equations using cubic B-spline approximation, Int J Comput Math, 91, 12, 2584-2602 (2014) · Zbl 1333.65078 · doi:10.1080/00207160.2014.884792 |
| [15] | Liu, J.; Gong, C.; Bao, W.; Tang, G.; Jiang, Y., Solving the Caputo fractional reaction-diffusion equation on GPU, Discrete Dyn Nat Soc (2014) · Zbl 1422.65164 · doi:10.1155/2014/820162 |
| [16] | Liu, Y.; Du, Y.; Li, H.; Wang, J., An \(H^1\)-Galerkin mixed finite element method for time fractional reaction-diffusion equation, J Appl Math Comput, 47, 103-117 (2015) · Zbl 1319.65097 · doi:10.1007/s12190-014-0764-7 |
| [17] | Mccartin, Bj, Theory of exponential splines, J Approx Theory, 66, 1, 1-23 (1991) · Zbl 0756.41019 · doi:10.1016/0021-9045(91)90050-K |
| [18] | Mohammadi, R., Exponential B-spline solution of convection-diffusion equations, Appl Math, 4, 6, 933-944 (2013) · doi:10.4236/am.2013.46129 |
| [19] | Mohammadi, R., Exponential B-spline collocation method for numerical solution of the generalized regularized long wave equation, Chin Phys B, 24, 5, 050206-910 (2015) · doi:10.1088/1674-1056/24/5/050206 |
| [20] | Oldham, Kb; Spanier, J., The fractional calculus: theory and applications of differentiation and integration to arbitrary order (1974), San Diego: Academic Press, San Diego · Zbl 0292.26011 |
| [21] | Podlubny, I., Fractional differential equations (1999), San Diego: Academic press, San Diego · Zbl 0924.34008 |
| [22] | Povstenko, Y., Linear fractional diffusion-wave equation for scientists and engineers (2015), New York: Birkhauser, New York · Zbl 1331.35004 · doi:10.1007/978-3-319-17954-4 |
| [23] | Rashidinia, J.; Mohmedi, E., Convergence analysis of tau scheme for the fractional reaction-diffusion equation, Eur Phys J Plus (2018) · doi:10.1140/epjp/i2018-12200-2 |
| [24] | Rida, Sz; El-Sayed, Ama; Arafa, Aam, On the solutions of time-fractional reaction-diffusion equations, Commun Nonlinear Sci Numer Simulat, 15, 12, 3847-3854 (2010) · Zbl 1222.65115 · doi:10.1016/j.cnsns.2010.02.007 |
| [25] | Turut, V.; Guzel, N., Comparing numerical methods for solving time-fractional reaction-diffusion equations, ISRN Math Anal 2012 (2012) · Zbl 1250.65128 · doi:10.5402/2012/737206 |
| [26] | Wang, Ql; Liu, J.; Gong, Cy; Tang, Xt; Fu, Gt; Xing, Zc, An efficient parallel algorithm for Caputo fractional reaction-diffusion equation with implicit finite-difference method, Adv Differ Equ, 1, 207-218 (2016) · Zbl 1419.34041 · doi:10.1186/s13662-016-0929-9 |
| [27] | Zhang, J.; Yang, X., A class of efficient difference method for time fractional reaction-diffusion equation, Comp Appl Math, 37, 4, 4376-4396 (2018) · Zbl 1404.65107 · doi:10.1007/s40314-018-0579-5 |
| [28] | Zhu, X.; Nie, Y.; Yuan, Z.; Wang, J.; Yang, Z., An exponential B-spline collocation method for the fractional sub-diffusion equation, Adv Differ Equ (2017) · Zbl 1444.65053 · doi:10.1186/s13662-017-1328-6 |
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