International Journal of Nonlinear Analysis and Applications
The numerical solution of time fractional generalized Benjamin-Bona-Mahony equation via the Sinc functions
Document Type : Research Paper
Author
Islamabad Faculty of Engineering, Razi University, Kermanshah, Iran
Abstract
The aim of this paper is to analyze a numerical method for solving the time fractional Benjamin-Bona-Mahony (BBM) equation. The time variable has been discretized by using the finite forward difference procedure. The unconditionally stable semi-discrete formula has been proven. Then we apply the Sinc collocation method to approximate the solution of the semi-discrete scheme. The exponential convergence rate of the Sinc method has also been proven. To show the efficiency of the proposed method, two examples were given. Numerical results verified the theoretical results and illustrate the efficiency and accuracy of the method compared with other methods.
Keywords
[1] A. Atabaigi, A. Barati and H. Norouzi, Bifurcation analysis of an enzyme-catalyzed reaction-diffusion system, Comput. Math. Appl. 75 (2018), 4361–4377.
[2] T. Ak, S. Dhawan, S.B.G. Karakoc, S.K. Bhowmik and K.R. Raslan,Numerical study of rosenau-KdV equation using finite element method based on collocation approach, Math. Model. Anal. 22 (2017), no. 3, 373–388.
[3] A. Barati and A. Atabaigi, Numerical solutions of a system of singularly perturbed reaction-diffusion problems, Filomat 33 (2019), no. 15, 4889–4905.
[4] T.B. Benjamin, J.L. Bona and J.J. Mahony,Model equations for long waves in nonlinear dispersive systems, Philos. Trans.R. Soc. Lond. Ser. A. 272 (1972), 47–78.
[5] B. Bialecki, Sinc-collocation methods for two-point boundary value problems, IMA J. Numer. Anal. 11 (1991), 357–375.
[6] A. Bueno-Orovio, D. Kay, V. Grau,B. Rodriguez and K. Burrage,Fractional diffusion models of cardiac electrical propagation: role of structural heterogeneity in dispersion of repolarization, J.R. Soc. Interface 11 (2014), 20140352.
[7] K. Burrage,N. Hale, and D. Kay,An efficient implicit FEM scheme for fractional-in-space reaction-diffusion equations, SIAM J. Sci. Comput. 34 (2012), A2145–A2172.
[8] I. Dag and M.N. Ozer, ¨ Approximation of RLW equation by least square cubic B-spline finite element method, Appl. Math. Model. 25 (2001), 221–231.
[9] I. Dag, B. Saka and D. Irk, Application of cubic B-splines for numerical solution of the RLW equation, Appl. Math. Comput. 159 (2004), 373–389.
[10] A. Esen and S. Kutluay, Application of a lumped Galerkin method to the regularized long wave equation, Appl. Math. Comput. 174 (2006), 833–845.
[11] R. FitzHugh,Mathematical models of threshold phenomena in the nerve membrane, Bull. Math. Biophys. 17 (1955), 257–278.
[12] L. R.T. Gardner, G.A. Gardner and I. Dag,A B-spline finite element method for the regularized long wave equation, Commun. Numer. Methods Eng. 11 (1995), 59–68.
[13] R. Herrmann,Fractional calculus: An introduction for physicists, World Scientific. Singapore, 2011.
[14] A. Hasegawa, and K. Mima, Pseudo three-dimensional turbulence in magnetized nonuniform plasma, Phys. Fluids 21 (1978), 87–92.
[15] J.W. Horton ,Fluctuation spectra of a drift wave soliton gas, Phys. Fluids 25(1982), 1838–1843.
[2] T. Ak, S. Dhawan, S.B.G. Karakoc, S.K. Bhowmik and K.R. Raslan,Numerical study of rosenau-KdV equation using finite element method based on collocation approach, Math. Model. Anal. 22 (2017), no. 3, 373–388.
[3] A. Barati and A. Atabaigi, Numerical solutions of a system of singularly perturbed reaction-diffusion problems, Filomat 33 (2019), no. 15, 4889–4905.
[4] T.B. Benjamin, J.L. Bona and J.J. Mahony,Model equations for long waves in nonlinear dispersive systems, Philos. Trans.R. Soc. Lond. Ser. A. 272 (1972), 47–78.
[5] B. Bialecki, Sinc-collocation methods for two-point boundary value problems, IMA J. Numer. Anal. 11 (1991), 357–375.
[6] A. Bueno-Orovio, D. Kay, V. Grau,B. Rodriguez and K. Burrage,Fractional diffusion models of cardiac electrical propagation: role of structural heterogeneity in dispersion of repolarization, J.R. Soc. Interface 11 (2014), 20140352.
[7] K. Burrage,N. Hale, and D. Kay,An efficient implicit FEM scheme for fractional-in-space reaction-diffusion equations, SIAM J. Sci. Comput. 34 (2012), A2145–A2172.
[8] I. Dag and M.N. Ozer, ¨ Approximation of RLW equation by least square cubic B-spline finite element method, Appl. Math. Model. 25 (2001), 221–231.
[9] I. Dag, B. Saka and D. Irk, Application of cubic B-splines for numerical solution of the RLW equation, Appl. Math. Comput. 159 (2004), 373–389.
[10] A. Esen and S. Kutluay, Application of a lumped Galerkin method to the regularized long wave equation, Appl. Math. Comput. 174 (2006), 833–845.
[11] R. FitzHugh,Mathematical models of threshold phenomena in the nerve membrane, Bull. Math. Biophys. 17 (1955), 257–278.
[12] L. R.T. Gardner, G.A. Gardner and I. Dag,A B-spline finite element method for the regularized long wave equation, Commun. Numer. Methods Eng. 11 (1995), 59–68.
[13] R. Herrmann,Fractional calculus: An introduction for physicists, World Scientific. Singapore, 2011.
[14] A. Hasegawa, and K. Mima, Pseudo three-dimensional turbulence in magnetized nonuniform plasma, Phys. Fluids 21 (1978), 87–92.
[15] J.W. Horton ,Fluctuation spectra of a drift wave soliton gas, Phys. Fluids 25(1982), 1838–1843.
[16] S. Islam, S. Haq and A. Ali, A meshfree method for the numerical solution of RLW equation, J. Comput. Appl. Math. 223 (2009), 997–1012.
[17] D. Kaya, A numerical simulation of solitary-wave solutions of the generalized regularized long-wave equation, Appl. Math. Comput. 149 (2004), 833–841.
[18] Y. Luchko and A. Punzi, Modeling anomalous heat transport in geothermal reservoirs via fractional diffusion equations, Int. J. Geomath. 1 (2011), no. 1, 257–276.
[19] P. Lyu and S. Vong, A linearized second-order finite difference scheme for time fractional generalized BBM equation, Appl. Math. Lett. 78 (2018), 16–22.
[20] P. Lyu and S. Vong, A high-order method with a temporal nonuniform mesh for a time-fractional Benjamin BonaMahony equation, J. Sci. Comput. 80(2019), 1607–1628.
[21] J. Lund and K. Bowers, Sinc methods for quadrature and differential equations, SIAM, Philadelphia, PA. 1992.
[22] Y. M.Lin and C.J. Xu, Finite difference/spectral approximations for the time-fractional diffusion equation. J. Comput. Phys. 225 (2007), 1533–1552.
[23] R. Metzler and J. Klafter, The random walk’s guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep. 339 (2000), 1–77.
[24] M. Mirzazadeh, M. Ekici and A. Sonmezoglu, On the solutions of the space and time fractional Benjamin– Bona–Mahony equation, Iran. J. Sci. Technol. Trans. Sci. 41 (2017), 819–836.
[25] K.S. Miller and B. Ross, An introduction to the fractional calculus and fractional differential equations, Wiley, NewYork, 1993.
[26] I. Podlubny,Fractional differential equations, Academic Press, New York, 1999.
[27] J. Rashidinia, A. Barati and M. Nabati,Application of Sinc-Galerkin method to singularly perturbed parabolic convection-diffusion problems, Numer. Algor. 66 (2014), 643–662.
[28] W. Rudin, Principles of Mathematical Analysis, Third Edition. McGraw-Hill Inc., 1976.
[29] R.A. Saxton, Dynamic instability of the liquid crystal director, Contem. Math. 100 (1989), 325–330.
[30] F. Stenger, Numerical methods based on Sinc and analytic functions, Springer, New york, 1993.
[31] A. Saadatmandi, A. Asadi and A. Eftekhari, Collocation method using quintic B-spline and sinc functions for solving a model of squeezing flow between two infinite plates, Int. J. Comput. Math. 93 (2016), no. 11, 1921–1936.
[32] Y. M Wang, A high-order linearized and compact difference method for the time-fractional Benjamin–Bona–Mahony equation, Appl. Math. Lett. 105 (2020), 106339.
[33] M. Youssef and G. Baumann, Troesch’s problem solved by Sinc methods, Math. Comput. Simul. 162 (2019), 31–44.
[17] D. Kaya, A numerical simulation of solitary-wave solutions of the generalized regularized long-wave equation, Appl. Math. Comput. 149 (2004), 833–841.
[18] Y. Luchko and A. Punzi, Modeling anomalous heat transport in geothermal reservoirs via fractional diffusion equations, Int. J. Geomath. 1 (2011), no. 1, 257–276.
[19] P. Lyu and S. Vong, A linearized second-order finite difference scheme for time fractional generalized BBM equation, Appl. Math. Lett. 78 (2018), 16–22.
[20] P. Lyu and S. Vong, A high-order method with a temporal nonuniform mesh for a time-fractional Benjamin BonaMahony equation, J. Sci. Comput. 80(2019), 1607–1628.
[21] J. Lund and K. Bowers, Sinc methods for quadrature and differential equations, SIAM, Philadelphia, PA. 1992.
[22] Y. M.Lin and C.J. Xu, Finite difference/spectral approximations for the time-fractional diffusion equation. J. Comput. Phys. 225 (2007), 1533–1552.
[23] R. Metzler and J. Klafter, The random walk’s guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep. 339 (2000), 1–77.
[24] M. Mirzazadeh, M. Ekici and A. Sonmezoglu, On the solutions of the space and time fractional Benjamin– Bona–Mahony equation, Iran. J. Sci. Technol. Trans. Sci. 41 (2017), 819–836.
[25] K.S. Miller and B. Ross, An introduction to the fractional calculus and fractional differential equations, Wiley, NewYork, 1993.
[26] I. Podlubny,Fractional differential equations, Academic Press, New York, 1999.
[27] J. Rashidinia, A. Barati and M. Nabati,Application of Sinc-Galerkin method to singularly perturbed parabolic convection-diffusion problems, Numer. Algor. 66 (2014), 643–662.
[28] W. Rudin, Principles of Mathematical Analysis, Third Edition. McGraw-Hill Inc., 1976.
[29] R.A. Saxton, Dynamic instability of the liquid crystal director, Contem. Math. 100 (1989), 325–330.
[30] F. Stenger, Numerical methods based on Sinc and analytic functions, Springer, New york, 1993.
[31] A. Saadatmandi, A. Asadi and A. Eftekhari, Collocation method using quintic B-spline and sinc functions for solving a model of squeezing flow between two infinite plates, Int. J. Comput. Math. 93 (2016), no. 11, 1921–1936.
[32] Y. M Wang, A high-order linearized and compact difference method for the time-fractional Benjamin–Bona–Mahony equation, Appl. Math. Lett. 105 (2020), 106339.
[33] M. Youssef and G. Baumann, Troesch’s problem solved by Sinc methods, Math. Comput. Simul. 162 (2019), 31–44.
)
Volume 14, Issue 1
January 2023Pages 2759-2770
- Receive Date: 02 June 2021
- Revise Date: 08 August 2021
- Accept Date: 21 August 2021