Spectral method for one dimensional Benjamin-Bona-Mahony-Burgers equation using the transformed generalized Jacobi polynomial
Abstract
The Benjamin-Bona-Mahony-Burgers equation (BBMBE) plays a fundemental role in many application scenarios. In this paper, we study a spectral method for the BBMBE with homogeneous boundary conditions. We propose a spectral scheme using the transformed generalized Jacobi polynomial in combination of the explicit fourth-order Runge-Kutta method in time. The boundedness, the generalized stability and the convergence of the proposed scheme are proved. The extensive numerical examples show the efficiency of the new proposed scheme and coincide well with the theoretical analysis. The advantages of our new approach are as follows: (i) the use of the transformed generalized Jacobi polynomial simplifies the theoretical analysis and brings a sparse discrete system; (ii) the numerical solution is spectral accuracy in space.
Keyword : spectral method, Benjamin-Bona-Mahony-Burgers equation, generalized Jacobi function
How to Cite
Zhou, Y., & Jiao, Y. (2024). Spectral method for one dimensional Benjamin-Bona-Mahony-Burgers equation using the transformed generalized Jacobi polynomial. Mathematical Modelling and Analysis, 29(3), 509–524. https://doi.org/10.3846/mma.2024.18595
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
S. Abbasbandy and A. Shirzadi. The first integral method for modified BenjaminBona-Mahony equation. Commun. Nonlinear Sci. Numer. Simul., 15:1759–1764, 2010. https://doi.org/10.1016/j.cnsns.2009.08.003
M. Abdollahzadeh, M. Ghanbarpour and S. Kashani. Exact travelling solutions for Benjamin-Bona-Mahony-Burgers equations by (G’/G)expansion method. Int. J. Appl. Math. Comput., 3:70–76, 2011. https://doi.org/10.0000/IJAMC.2011.3.1.110
K. Al-Khaled, S. Momani and A. Alawneh. Approximate wave solutions for generalized Benjamin-Bona-Mahony-Burgers equations. Appl. Math. Comput., 171:281–292, 2005. https://doi.org/10.1016/j.amc.2005.01.056
D.N. Arnold, J. Douglas Jr. and V. Thomée. Superconvergence of finite element approximation to the solution of a Sobolev equation in a single space variable. Math. Comp., 27(153):737–743, 1981. https://doi.org/10.1090/S0025-5718-1981-0595041-4
S. Arora, R. Jain and V.K. Kukreja. Solution of Benjamin-Bona-Mahony-Burgers equation using collocation method with quintic Hermite splines. Appl. Numer. Math., 154:1–16, 2020. https://doi.org/10.1016/j.apnum.2020.03.015
M. Aslefallah, S. Abbasbandy and E. Shivanian. Meshless formulation to two-dimensional nonlinear problem of generalized Benjamin–Bona–Mahony–Burgers through singular boundary method: Analysis of stability and convergence. Numer. Meth. Partial Diff. Eq., 36:249–267, 2020. https://doi.org/10.1002/num.22426
G. Barenblatt, I. Zheltov and I. Kochina. Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks. J. Appl. Math. Mech., 24(5):852–864, 1960. https://doi.org/10.1016/0021-8928(60)90107-6
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, Math. Phys. Sci, 272(1220):47–48, 1972. https://doi.org/10.1098/rsta.1972.0032
J.M. Burgers. A mathematical model illustrating the theory of turbulence. in: Advances in Applied Mechanics, Academic Press, Inc., New York, pp. 171–199, 1948. https://doi.org/10.1016/S0065-2156(08)70100-5
M. Dehghan, M. Abbaszadeha and A. Mohebbib. The use of interpolating element-free Galerkin technique for solving 2D generalized Benjamin–Bona–Mahony–Burgers and regularized long-wave equations on nonrectangular domains with error estimate. J. Comput. Appl. Math., 286:211–231, 2015. https://doi.org/10.1016/j.cam.2015.03.012
S.A. El-Wakil, M.A. Abdou and A. Hendi. New periodic wave solutions via Exp-function method. Phys. Lett. A, 372:830–840, 2008. https://doi.org/10.1016/j.physleta.2007.08.033
R.E. Ewing. Time-stepping Galerkin methods for nonlinear Sobolev partial differential equations. SIAM J. Numer. Anal., 15(6):1125–1150, 1978. https://doi.org/10.1137/0715075
Z.Z. Ganji, D.D. Ganji and H. Bararnia. Approximate general and explicit solutions of nonlinear BBMB equations by Exp-Function method. Appl. Math. Model., 33:1836–1841, 2009. https://doi.org/10.1016/j.apm.2008.03.005
B.Y. Guo. Spectral methods and their applications. World Scientific, 1998.
B.Y. Guo and Y.J. Jiao. Spectral method for Navier-Stokes equations with slip boundary conditions. J. Sci. Comput., 58(1):249–274, 2014. https://doi.org/10.1007/s10915-013-9729-5
B.Y Guo, J. Shen and L.L. Wang. Generalized Jacobi polynomials/functions and their applications. Appl. Numer. Math., 59(5):1011–1028, 2009. https://doi.org/10.1016/j.apnum.2008.04.003
M. Izadi and M.E. Samei. Time accurate solution to Benjamin-Bona-MahonyBurgers equation via Taylor-Boubaker series scheme. Bound. Value Probl., 2022(1):17, 2022. https://doi.org/10.1186/s13661-022-01598-x
T. Kadri, N. Khiari, F. Abidi and K. Omrani. Methods for the numerical solution of the Benjamin-Bona-Mahony-Burgers equation. Numer. Meth. Partial Diff. Eq., 24(6):1501–1516, 2008. https://doi.org/10.1002/num.20330
S.B.G. Karakoc and K.K. Ali. Theoretical and computational structures on solitary wave solutions of Benjamin Bona Mahony-Burgers equation. Tbilisi Math. J., 14(2):33–50, 2021. https://doi.org/10.32513/tmj/19322008120
S.B.G. Karakoc and S.K. Bhowmik. Galerkin finite element solution for Benjamin-Bona-Mahony-Burgers equation with cubic B-splines. Comput. Math. Appl., 77(7):1917–1932, 2019. https://doi.org/10.1016/j.camwa.2018.11.023
M. Mei. Large-time behavior of solution for generalized BenjaminBona-Mahony-Burgers equations. Nonlinear Anal., 33:699–714, 1998. https://doi.org/10.1016/S0362-546X(97)00674-3
A. Mohebbi and Z. Faraz. Solitary wave solution of nonlinear Benjamin-BonaMahony-Burgers equation using a high-order difference scheme. Comput. Appl. Math., 36(2):915–927, 2017. https://doi.org/10.1007/s40314-015-0272-x
K. Omrani and M. Ayadi. Finite difference discretization of the Benjamin-BonaMahony-Burgers equation. Numer. Meth. Partial Diff. Eq., 24(1):239–248, 2008. https://doi.org/10.1002/num.20256
D.H. Peregrine. Calculations of the development of an undular bore. J. Fluid Mech., 25:321–330, 1966. https://doi.org/10.1017/S0022112066001678
C.A. Gómez S., A.H. Salas and B.A. Frias. New periodic and soliton solutions for the Generalized BBM and Burgers–BBM equations. Appl. Math. Comput., 217(4):1430–1434, 2010. https://doi.org/10.1016/j.amc.2009.05.068
T.W. Ting. Certain non-steady flows of second-order fluids. Arch. Ration. Mech. Anal., 14(1):1–26, 1963. https://doi.org/10.1007/BF00250690
Q.H. Xiao and Z.Z. Chen. Degenerate boundary layer solutions to the generalized Benjamin–Bona–Mahony–Burgers equation. Acta Math. Sci., 32B(5):1743– 1758, 2012. https://doi.org/10.1016/S0252-9602(12)60138-6
H. Yin and J. Hu. Exponential decay rate of solutions toward traveling waves for the Cauchy problem of generalized Benjamin–Bona–Mahony–Burgers equations. Nonlinear Anal., 73:1729–1738, 2010. https://doi.org/10.1016/j.na.2010.04.078
M. Zarebnia and R. Parvaz. On the numerical treatment and analysis of Benjamin–Bona–Mahony–Burgers equation. Appl. Math. Comput., 284:79–88, 2016. https://doi.org/10.1016/j.amc.2016.02.037
Q.F. Zhang, L.L. Liu and J.Y. Zhang. The numerical analysis of two linearized difference schemes for the Benjamin–Bona–Mahony–Burgers equation. Numer. Meth. Partial Diff. Eq., 36:1790–1810, 2020. https://doi.org/10.1002/num.22504
M. Abdollahzadeh, M. Ghanbarpour and S. Kashani. Exact travelling solutions for Benjamin-Bona-Mahony-Burgers equations by (G’/G)expansion method. Int. J. Appl. Math. Comput., 3:70–76, 2011. https://doi.org/10.0000/IJAMC.2011.3.1.110
K. Al-Khaled, S. Momani and A. Alawneh. Approximate wave solutions for generalized Benjamin-Bona-Mahony-Burgers equations. Appl. Math. Comput., 171:281–292, 2005. https://doi.org/10.1016/j.amc.2005.01.056
D.N. Arnold, J. Douglas Jr. and V. Thomée. Superconvergence of finite element approximation to the solution of a Sobolev equation in a single space variable. Math. Comp., 27(153):737–743, 1981. https://doi.org/10.1090/S0025-5718-1981-0595041-4
S. Arora, R. Jain and V.K. Kukreja. Solution of Benjamin-Bona-Mahony-Burgers equation using collocation method with quintic Hermite splines. Appl. Numer. Math., 154:1–16, 2020. https://doi.org/10.1016/j.apnum.2020.03.015
M. Aslefallah, S. Abbasbandy and E. Shivanian. Meshless formulation to two-dimensional nonlinear problem of generalized Benjamin–Bona–Mahony–Burgers through singular boundary method: Analysis of stability and convergence. Numer. Meth. Partial Diff. Eq., 36:249–267, 2020. https://doi.org/10.1002/num.22426
G. Barenblatt, I. Zheltov and I. Kochina. Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks. J. Appl. Math. Mech., 24(5):852–864, 1960. https://doi.org/10.1016/0021-8928(60)90107-6
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, Math. Phys. Sci, 272(1220):47–48, 1972. https://doi.org/10.1098/rsta.1972.0032
J.M. Burgers. A mathematical model illustrating the theory of turbulence. in: Advances in Applied Mechanics, Academic Press, Inc., New York, pp. 171–199, 1948. https://doi.org/10.1016/S0065-2156(08)70100-5
M. Dehghan, M. Abbaszadeha and A. Mohebbib. The use of interpolating element-free Galerkin technique for solving 2D generalized Benjamin–Bona–Mahony–Burgers and regularized long-wave equations on nonrectangular domains with error estimate. J. Comput. Appl. Math., 286:211–231, 2015. https://doi.org/10.1016/j.cam.2015.03.012
S.A. El-Wakil, M.A. Abdou and A. Hendi. New periodic wave solutions via Exp-function method. Phys. Lett. A, 372:830–840, 2008. https://doi.org/10.1016/j.physleta.2007.08.033
R.E. Ewing. Time-stepping Galerkin methods for nonlinear Sobolev partial differential equations. SIAM J. Numer. Anal., 15(6):1125–1150, 1978. https://doi.org/10.1137/0715075
Z.Z. Ganji, D.D. Ganji and H. Bararnia. Approximate general and explicit solutions of nonlinear BBMB equations by Exp-Function method. Appl. Math. Model., 33:1836–1841, 2009. https://doi.org/10.1016/j.apm.2008.03.005
B.Y. Guo. Spectral methods and their applications. World Scientific, 1998.
B.Y. Guo and Y.J. Jiao. Spectral method for Navier-Stokes equations with slip boundary conditions. J. Sci. Comput., 58(1):249–274, 2014. https://doi.org/10.1007/s10915-013-9729-5
B.Y Guo, J. Shen and L.L. Wang. Generalized Jacobi polynomials/functions and their applications. Appl. Numer. Math., 59(5):1011–1028, 2009. https://doi.org/10.1016/j.apnum.2008.04.003
M. Izadi and M.E. Samei. Time accurate solution to Benjamin-Bona-MahonyBurgers equation via Taylor-Boubaker series scheme. Bound. Value Probl., 2022(1):17, 2022. https://doi.org/10.1186/s13661-022-01598-x
T. Kadri, N. Khiari, F. Abidi and K. Omrani. Methods for the numerical solution of the Benjamin-Bona-Mahony-Burgers equation. Numer. Meth. Partial Diff. Eq., 24(6):1501–1516, 2008. https://doi.org/10.1002/num.20330
S.B.G. Karakoc and K.K. Ali. Theoretical and computational structures on solitary wave solutions of Benjamin Bona Mahony-Burgers equation. Tbilisi Math. J., 14(2):33–50, 2021. https://doi.org/10.32513/tmj/19322008120
S.B.G. Karakoc and S.K. Bhowmik. Galerkin finite element solution for Benjamin-Bona-Mahony-Burgers equation with cubic B-splines. Comput. Math. Appl., 77(7):1917–1932, 2019. https://doi.org/10.1016/j.camwa.2018.11.023
M. Mei. Large-time behavior of solution for generalized BenjaminBona-Mahony-Burgers equations. Nonlinear Anal., 33:699–714, 1998. https://doi.org/10.1016/S0362-546X(97)00674-3
A. Mohebbi and Z. Faraz. Solitary wave solution of nonlinear Benjamin-BonaMahony-Burgers equation using a high-order difference scheme. Comput. Appl. Math., 36(2):915–927, 2017. https://doi.org/10.1007/s40314-015-0272-x
K. Omrani and M. Ayadi. Finite difference discretization of the Benjamin-BonaMahony-Burgers equation. Numer. Meth. Partial Diff. Eq., 24(1):239–248, 2008. https://doi.org/10.1002/num.20256
D.H. Peregrine. Calculations of the development of an undular bore. J. Fluid Mech., 25:321–330, 1966. https://doi.org/10.1017/S0022112066001678
C.A. Gómez S., A.H. Salas and B.A. Frias. New periodic and soliton solutions for the Generalized BBM and Burgers–BBM equations. Appl. Math. Comput., 217(4):1430–1434, 2010. https://doi.org/10.1016/j.amc.2009.05.068
T.W. Ting. Certain non-steady flows of second-order fluids. Arch. Ration. Mech. Anal., 14(1):1–26, 1963. https://doi.org/10.1007/BF00250690
Q.H. Xiao and Z.Z. Chen. Degenerate boundary layer solutions to the generalized Benjamin–Bona–Mahony–Burgers equation. Acta Math. Sci., 32B(5):1743– 1758, 2012. https://doi.org/10.1016/S0252-9602(12)60138-6
H. Yin and J. Hu. Exponential decay rate of solutions toward traveling waves for the Cauchy problem of generalized Benjamin–Bona–Mahony–Burgers equations. Nonlinear Anal., 73:1729–1738, 2010. https://doi.org/10.1016/j.na.2010.04.078
M. Zarebnia and R. Parvaz. On the numerical treatment and analysis of Benjamin–Bona–Mahony–Burgers equation. Appl. Math. Comput., 284:79–88, 2016. https://doi.org/10.1016/j.amc.2016.02.037
Q.F. Zhang, L.L. Liu and J.Y. Zhang. The numerical analysis of two linearized difference schemes for the Benjamin–Bona–Mahony–Burgers equation. Numer. Meth. Partial Diff. Eq., 36:1790–1810, 2020. https://doi.org/10.1002/num.22504