References
- S. E. Esipov, “Coupled Burgers equations: a model of polydispersive sedimentation,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics, vol. 52, no. 4, pp. 3711–3718, 1995. DOI: https://doi.org/10.1103/physreve.52.3711.
- J. Nee and J. Duan, “Limit set of trajectories of the coupled viscous Burgers' equations,” Appl. Math. Lett., vol. 11, no. 1, pp. 57–61, 1998. DOI: https://doi.org/10.1016/S0893-9659(97)00133-X.
- J. M. Burgers, “A mathematical model illustrating the theory of turbulence,” in Advances in Applied Mechanics, vol. 1. Elsevier, 1948, pp. 171–199.
- J. D. Cole, “On a quasi-linear parabolic equation occurring in aerodynamics,” Quart. Appl. Math., vol. 9, no. 3, pp. 225–236, 1951. DOI: https://doi.org/10.1090/qam/42889.
- D. Kaya, “An explicit solution of coupled viscous Burgers' equation by the decomposition method,” Int. J. Mathe. Math. Sci., vol. 27, no. 11, pp. 675–680, 2001. DOI: https://doi.org/10.1155/S0161171201010249.
- A. H. Khater, R. S. Temsah, and M. M. Hassan, “A Chebyshev spectral collocation method for solving Burgers’-type equations,” J. Comput. Appl. Math., vol. 222, no. 2, pp. 333–350, 2008. DOI: https://doi.org/10.1016/j.cam.2007.11.007.
- M. Dehghan, A. Hamidi, and M. Shakourifar, “The solution of coupled Burgers’ equations using Adomian–Pade technique,” Appl. Math. Comput., vol. 189, no. 2, pp. 1034–1047, 2007. DOI: https://doi.org/10.1016/j.amc.2006.11.179.
- V. K. Srivastava, M. K. Awasthi, and M. Tamsir, “A fully implicit finite-difference solution to one dimensional coupled nonlinear Burgers’ equations,” Int. J. Math. Sci., vol. 7, no. 4, pp. 23, 2013.
- V. K. Srivastava, M. Tamsir, M. K. Awasthi, and S. Singh, “One-dimensional coupled Burgers’ equation and its numerical solution by an implicit logarithmic finite-difference method,” AIP Adv., vol. 4, no. 3, pp. 037119, 2014. DOI: https://doi.org/10.1063/1.4869637.
- R. C. Mittal and R. Jiwari, “A differential quadrature method for numerical solutions of Burgers'‐type equations,” Int. J. Numer. Methods Heat Fluid Flow, vol. 22, no. 7, pp. 880–895, 2012. DOI: https://doi.org/10.1108/09615531211255761.
- R. C. Mittal and G. Arora, “Numerical solution of the coupled viscous Burgers’ equation,” Commun. Nonlinear Sci. Numer. Simul., vol. 16, no. 3, pp. 1304–1313, 2011. DOI: https://doi.org/10.1016/j.cnsns.2010.06.028.
- A. Rashid and A. I. B. M. Ismail, “A Fourier pseudo-spectral method for solving coupled viscous Burgers equations,” Comput. Methods Appl. Math., vol. 9, no. 4, pp. 412–420, 2009. DOI: https://doi.org/10.2478/cmam-2009-0026.
- R. Mokhtari, A. S. Toodar, and N. G. Chegini, “Application of the generalized differential quadrature method in solving Burgers' equations,” Commun. Theor. Phys., vol. 56, no. 6, pp. 1009–1015, 2011. DOI: https://doi.org/10.1088/0253-6102/56/6/06.
- H. Lai and C. Ma, “A new lattice Boltzmann model for solving the coupled viscous Burgers’ equation,” Phys. A, vol. 395, pp. 445–457, 2014. DOI: https://doi.org/10.1016/j.physa.2013.10.030.
- V. K. Srivastava, M. K. Awasthi, M. Tamsir, and S. Singh, “An implicit finite-difference solution to one-dimensional coupled Burgers' equations,” Asian-Eur. J. Math., vol. 06, no. 04, pp. 1350058, 2013. DOI: https://doi.org/10.1142/S1793557113500587.
- H. M. Salih, L. N. M. Tawfiq, and Z. R. Yahya, “Numerical solution of the coupled viscous Burgers’ equation via cubic trigonometric B-spline approach,” Math. Stat. J., vol. 2, no. 11, 2016.
- K. R. Raslan, T. S. El-Danaf, and K. K. Ali, “Collocation method with quintic B-spline method for solving coupled Burgers' equations,” FJAM, vol. 96, no. 1, pp. 55–75, 2017. DOI: https://doi.org/10.17654/AM096010055.
- F. Liu, Y. Wang, and S. Li, “Barycentric interpolation collocation method for solving the coupled viscous Burgers' equations,” Int. J. Comput. Math., vol. 95, no. 11, pp. 2162–2173, 2018. DOI: https://doi.org/10.1080/00207160.2017.1384546.
- Q. Li, Z. Chai, and B. Shi, “A novel lattice Boltzmann model for the coupled viscous Burgers’ equations,” Appl. Math. Comput., vol. 250, pp. 948–957, 2015. DOI: https://doi.org/10.1016/j.amc.2014.11.036.
- H. P. Bhatt and A. Q. M. Khaliq, “Fourth-order compact schemes for the numerical simulation of coupled Burgers’ equation,” Comput. Phys. Commun., vol. 200, pp. 117–138, 2016. DOI: https://doi.org/10.1016/j.cpc.2015.11.007.
- R. Bellman, B. G. Kashef, and J. Casti, “Differential quadrature: a technique for the rapid solution of non-linear partial differential equations,” J. Comput. Phys., vol. 10, no. 1, pp. 40–52, 1972. DOI: https://doi.org/10.1016/0021-9991(72)90089-7.
- J. R. Quan and C. T. Chang, “New insights in solving distributed system equations by the quadrature methods-I,” Comput. Chem. Eng., vol. 13, no. 7, pp. 779–788, 1989. DOI: https://doi.org/10.1016/0098-1354(89)85051-3.
- J. R. Quan and C. T. Chang, “New insights in solving distributed system equations by the quadrature methods-II,” Comput. Chem. Eng., vol. 13, no. 9, pp. 1017–1024, 1989. DOI: https://doi.org/10.1016/0098-1354(89)87043-7.
- C. Shu and B. E. Richards, “High resolution of natural convection in a square cavity by generalized differential quadrature,” Proceedings of Third Conference on Adv. Numer. Methods Eng. Theory Appl, Swansea, UK, 1990, vol. 2, pp. 978–985.
- C. Shu, “Generalized differential-integral quadrature and application to the simulation of imcompressible viscous flows including parallel computation,” Ph.D. thesis, Univ. of Glasgow, UK, 1991.
- C. Shu, Differential Quadrature and Its Application in Engineering. Springer Science & Business Media, 2000.
- G. Arora and B. K. Singh, “Numerical solution of Burgers’ equation with modified cubic B-spline differential quadrature method,” Appl. Math. Comput., vol. 224, pp. 166–177, 2013. DOI: https://doi.org/10.1016/j.amc.2013.08.071.
- H. S. Shukla, M. Tamsir, R. Jiwari, and V. K. Srivastava, “A numerical algorithm for computation modelling of 3D non-linear wave equations based on exponential modified cubic B-spline differential quadrature method,” Int. J. Comput. Math., vol. 95, no. 4, pp. 752–766, 2018. DOI: https://doi.org/10.1080/00207160.2017.1296573.
- A. Korkmaz and İ. Dağ, “A differential quadrature algorithm for non-linear Schrödinger equation,” Nonlinear Dyn., vol. 56, no. 1-2, pp. 69–83, 2009. DOI: https://doi.org/10.1007/s11071-008-9380-0.
- A. Verma, R. Jiwari, and S. Kumar, “A numerical scheme based on differential quadrature method for numerical simulation of non-linear Klein-Gordon equation,” Int. J. Numer. Methods Heat Fluid Flow, vol. 24, no. 7, pp. 1390–1404, 2014. DOI: https://doi.org/10.1108/HFF-01-2013-0014.
- R. Jiwari, S. Pandit, and R. C. Mittal, “Numerical simulation of two-dimensional sine-Gordon solitons by differential quadrature method,” Comput. Phys. Commun., vol. 183, no. 3, pp. 600–616, 2012. DOI: https://doi.org/10.1016/j.cpc.2011.12.004.
- A. Korkmaz and İ. Dağ, “Cubic B‐spline differential quadrature methods for the advection‐diffusion equation,” Int. J. Numer. Methods Heat Fluid Flow, vol. 22, no. 8, pp. 1021–1036, 2012. DOI: https://doi.org/10.1108/09615531211271844.
- R. C. Mittal and S. Dahiya, “Numerical simulation of three-dimensional telegraphic equation using cubic B-spline differential quadrature method,” Appl. Math. Comput., vol. 313, pp. 442–452, 2017. DOI: https://doi.org/10.1016/j.amc.2017.06.015.
- M. Ghasemi, “High order approximations using spline-based differential quadrature method: implementation to the multi-dimensional PDEs,” Appl. Math. Model., vol. 46, pp. 63–80, 2017. DOI: https://doi.org/10.1016/j.apm.2017.01.052.
- R. Jiwari, S. Pandit, and R. C. Mittal, “A differential quadrature algorithm to solve the two dimensional linear hyperbolic telegraph equation with Dirichlet and Neumann boundary conditions,” Appl. Math. Comput., vol. 218, no. 13, pp. 7279–7294, 2012. DOI: https://doi.org/10.1016/j.amc.2012.01.006.
- A. Korkmaz and İ. Dağ, İ. “Solitary wave simulations of complex modified Korteweg–de Vries equation using differential quadrature method,” Comput. Phys. Commun., vol. 180, no. 9, pp. 1516–1523, 2009. DOI: https://doi.org/10.1016/j.cpc.2009.04.012.
- R. Jiwari, R. C. Mittal, and K. K. Sharma, “A numerical scheme based on weighted average differential quadrature method for the numerical solution of Burgers’ equation,” Appl. Math. Comput., vol. 219, no. 12, pp. 6680–6691, 2013. DOI: https://doi.org/10.1016/j.amc.2012.12.035.
- M. Tamsir, N. Dhiman, A. Chauhan, and A. Chauhan, “Solution of parabolic PDEs by modified quintic B-spline Crank-Nicolson collocation method,” Ain Shams Eng. J., vol. 12, no. 2, pp. 2073–2082, 2021. DOI: https://doi.org/10.1016/j.asej.2020.08.028.
- M. Tamsir and N. Dhiman, “DQM based on the modified form of CTB shape functions for coupled Burgers’ equation in 2D and 3D,” Int. J. Math. Eng. Manage. Sci., vol. 4, no. 4, pp. 1051–1067, 2019.
- M. Tamsir, V. K. Srivastava, and R. Jiwari, “An algorithm based on exponential modified cubic B-spline differential quadrature method for nonlinear Burgers’ equation,” Appl. Math. Comput., vol. 290, pp. 111–124, 2016. DOI: https://doi.org/10.1016/j.amc.2016.05.048.
- H. S. Shukla, M. Tamsir, V. K. Srivastava, and M. M. Rashidi, “Modified cubic B-spline differential quadrature method for numerical solution of three-dimensional coupled viscous Burger equation,” Mod. Phys. Lett. B, vol. 30, no. 11, pp. 1650110, 2016. DOI: https://doi.org/10.1142/S0217984916501104.
- M. Tamsir, N. Dhiman, and V. K. Srivastava, “Extended modified cubic B-spline algorithm for nonlinear Burgers' equation,” Beni-Suef Univ. J. Basic Appl. Sci., vol. 5, no. 3, pp. 244–254, 2016. DOI: https://doi.org/10.1016/j.bjbas.2016.09.001.
- H. S. Shukla and M. Tamsir, “Extended modified cubic B-spline algorithm for nonlinear Fisher’s reaction-diffusion equation,” Alex. Eng. J., vol. 55, no. 3, pp. 2871–2879, 2016. DOI: https://doi.org/10.1016/j.aej.2016.06.031.
- H. S. Shukla, M. Tamsir, V. K. Srivastava, and J. Kumar, “Numerical solution of two dimensional coupled viscous Burger equation using modified cubic B-spline differential quadrature method,” AIP Adv., vol. 4, no. 11, pp. 117134, 2014. DOI: https://doi.org/10.1063/1.4902507.
- V. K. Srivastava, M. T. Ashutosh and M. Tamsir, “Generating exact solution of three-dimensional coupled unsteady nonlinear generalized viscous Burgers’ equations,” Int. J. Math. Sci., vol. 5, no. 1, pp. 1–13, 2013.
- J. P. Berrut and L. N. Trefethen, “Barycentric lagrange interpolation,” SIAM Rev., vol. 46, no. 3, pp. 501–517, 2004. DOI: https://doi.org/10.1137/S0036144502417715.
- N. J. Higham, “The numerical stability of barycentric Lagrange interpolation,” IMA J. Numer. Anal., vol. 24, no. 4, pp. 547–556, 2004. DOI: https://doi.org/10.1093/imanum/24.4.547.
- W. F. Mascarenhas, “The stability of barycentric interpolation at the Chebyshev points of the second kind,” Numer. Math., vol. 128, no. 2, pp. 265–300, 2014. DOI: https://doi.org/10.1007/s00211-014-0612-6.
- W. Mascarenhas and A. Camargo, “On the backward stability of the second barycentric formula for interpolation,” Dolomites Res. Notes Approx., vol. 7, no. 1, 2014.
- J. P. Berrut and G. Klein, “Recent advances in linear barycentric rational interpolation,” J. Comput. Appl. Math., vol. 259, pp. 95–107, 2014. DOI: https://doi.org/10.1016/j.cam.2013.03.044.
- S. Güttel and G. Klein, “Convergence of linear barycentric rational interpolation for analytic functions,” SIAM J. Numer. Anal., vol. 50, no. 5, pp. 2560–2580, 2012. DOI: https://doi.org/10.1137/120864787.
- W. Ma, B. Zhang, and H. Ma, “A meshless collocation approach with barycentric rational interpolation for two-dimensional hyperbolic telegraph equation,” Appl. Math. Comput., vol. 279, pp. 236–248, 2016. DOI: https://doi.org/10.1016/j.amc.2016.01.022.
- H. Liu, J. Huang, Y. Pan, and J. Zhang, “Barycentric interpolation collocation methods for solving linear and non-linear high-dimensional Fredholm integral equations,” J. Comput. Appl. Math., vol. 327, pp. 141–154, 2018. DOI: https://doi.org/10.1016/j.cam.2017.06.004.
- H. Liu, J. Huang, W. Zhang, and Y. Ma, “Meshfree approach for solving multi-dimensional systems of Fredholm integral equations via barycentric Lagrange interpolation,” Appl. Math. Comput., vol. 346, pp. 295–304, 2019. DOI: https://doi.org/10.1016/j.amc.2018.10.024.
- J. P. Berrut and H. D. Mittelmann, “Matrices for the direct determination of the barycentric weights of rational interpolation,” J. Comput. Appl. Math., vol. 78, no. 2, pp. 355–370, 1997. DOI: https://doi.org/10.1016/S0377-0427(96)00163-X.
- P. W. Lawrence and R. M. Corless, “Stability of rootfinding for barycentric Lagrange interpolants,” Numer. Algor., vol. 65, no. 3, pp. 447–464, 2014. DOI: https://doi.org/10.1007/s11075-013-9770-3.
- W. H. Luo, T. Z. Huang, X. M. Gu, and Y. Liu, “Barycentric rational collocation methods for a class of non-linear parabolic partial differential equations,” Appl. Math. Lett., vol. 68, pp. 13–19, 2017. DOI: https://doi.org/10.1016/j.aml.2016.12.011.
- M. S. Floater and K. Hormann, “Barycentric rational interpolation with no poles and high rates of approximation,” Numer. Math., vol. 107, no. 2, pp. 315–331, 2007. DOI: https://doi.org/10.1007/s00211-007-0093-y.
- H. Wu, Y. Wang, and W. Zhang, “Numerical solution of a class of non-linear partial differential equations by using barycentric interpolation collocation method,” Math. Probl. Eng., vol. 2018, pp. 1–10, 2018. DOI: https://doi.org/10.1155/2018/7260346.
- R. Abazari and A. Borhanifar, “Numerical study of the solution of the Burgers and coupled Burgers equations by a differential transformation method,” Comput. Math. Appl., vol. 59, no. 8, pp. 2711–2722, 2010. DOI: https://doi.org/10.1016/j.camwa.2010.01.039.
- Z. Rong-Pei, Y. Xi-Jun, and Z. Guo-Zhong, “Local discontinuous Galerkin method for solving Burgers and coupled Burgers equations,” Chin. Phys. B, vol. 20, no. 11, pp. 110205, 2011.
- A. Iserles, “Numerical solution of differential equations, by MK Jain. Pp 698.£17· 95. 1984. ISBN 0-85226-432-1 (Wiley Eastern),” Math. Gazette, vol. 69, no. 449, pp. 236–237, 1985.
- G. Arora and V. Joshi, “A computational approach using modified trigonometric cubic B-spline for numerical solution of Burgers’ equation in one and two dimensions,” Alex. Eng. J., vol. 57, no. 2, pp. 1087–1098, 2018. DOI: https://doi.org/10.1016/j.aej.2017.02.017.
- G. Arora, V. Joshi, and R. C. Mittal, “Numerical simulation of nonlinear Schrödinger equation in one and two dimensions,” Math. Models Comput. Simul., vol. 11, no. 4, pp. 634–648, 2019. DOI: https://doi.org/10.1134/S2070048219040070.