A multi-domain Galerkin method with numerical integration for the Burgers equation

References

• W.M. Abd-Elhameed and A.M. Alkenedri, New formulas for the repeated integrals of some Jacobi polynomials: Spectral solutions of even-order boundary value problems, Int. J. Appl. Comput. Math. 7 (2021), pp. 1–24.
   Google Scholar
• E.N. Aksan, A numerical solution of Burgers' equation by finite element method constructed on the method of discretization in time, Appl. Math. Comput. 170 (2005), pp. 895–904.
   Web of Science ®Google Scholar
• E.N. Aksan, Quadratic B-spline finite element method for numerical solution of the Burgers' equation, Appl. Math. Comput. 174 (2006), pp. 884–896.
   Web of Science ®Google Scholar
• I. Dağ, D. Irk, and B. Saka, A numerical solution of the Burgers' equation using cubic B-splines, Appl. Math. Comput. 163 (2005), pp. 199–211.
   Web of Science ®Google Scholar
• E.H. Doha, W.M. Abd-Elhameed, and A.H. Bhrawy, New spectral-Galerkin algorithms for direct solution of high even-order differential equations using symmetric generalized Jacobi polynomials, Collect. Math. 64 (2013), pp. 373–394.
   Web of Science ®Google Scholar
• P.G. Drazin and R.S. Jonson, Soliton: An Introduction, Cambridge University Press, New York, 1993.
   Google Scholar
• B.Y. Guo, Generalized stability of discretization and its applications to numerical solutions of nonlinear differential equations, Contemp. Math. 163 (1994), pp. 33–54.
   Google Scholar
• B.Y. Guo, Error estimation of Hermite spectral method for nonlinear partial differential equations, Math. Comput. 68 (1999), pp. 1069–1078.
   Web of Science ®Google Scholar
• B.Y. Guo and T.J. Wang, Composite generalized Laguerre-Legendre spectral method with domain decomposition and its application to Fokker-Planck equation in an infinite channel, Math. Comput. 78 (2009), pp. 129–151.
   Web of Science ®Google Scholar
• B.Y. Guo and X.H. Yu, Composite spectral method for exterior problems with polygonal obstacles, J. Sci. Comput. 59 (2014), pp. 439–472.
   Web of Science ®Google Scholar
• A. Jawad, M.D. Petkovic, and A. Biswas, Soliton solutions of Burgers equations and perturbed Burgers equation, Appl. Math. Comput. 216 (2010), pp. 3370–3377.
   Web of Science ®Google Scholar
• M. Landajuela, Burgers equation, Basque Center for Applied Mathematics, 2011.
   Google Scholar
• D. Liu, T.J. Wang, and C. Zhang, A fully discrete multi-domain spectral method for the Burgers equation on the whole line, submitted.
   Google Scholar
• J.D. Logan, Nonlinear Partial Differential Equations, Wiley, New York, 1994.
   Google Scholar
• H.P. Ma and B.Y. Guo, Composite Legendre-Laguerre pseudospectral approximation in unbounded domains, IMA J. Numer. Anal. 21 (2001), pp. 587–602.
   Web of Science ®Google Scholar
• Y. Maday and A. Quarteroni, Legendre and Chebyshev spectral approximations of Burgers' equation, Numer. Math. 37 (1981), pp. 321–332.
   Web of Science ®Google Scholar
• K. Pandey, L. Verma, and A.K. Verma, On a finite difference scheme for Burgers' equation, Appl. Math. Comput. 215 (2009), pp. 2206–2214.
   Web of Science ®Google Scholar
• M.M. Rashidi, G. Domairry, and S. Dinarvand, Approximate solutions for the Burger and regularized long wave equations by means of the homotopy analysis method, Commun. Nonlinear. Sci. Numer. Simulat. 14 (2009), pp. 708–717.
   Web of Science ®Google Scholar
• J. Shen, T. Tang, and L.L. Wang, Spectral Methods: Algorithms, Analysis and Applications, Springer Series in Computational Mathematics Vol. 41, Springer-Verlag, Berlin, Heidelberg, 2011.
   Google Scholar
• C.T. Sheng, Z.Q. Wang, and B.Y. Guo, A multistep Legendre-Gauss spectral collocation method for nonlinear Volterra integral equations, SIAM J. Numer. Anal. 52 (2014), pp. 1953–1980.
   Web of Science ®Google Scholar
• B.F. Smith, Domain decomposition methods for partial differential equations, Math. Comput. 53 (1997), pp. 225–243.
   Google Scholar
• T. Sun and T.J. Wang, Multi-domain decomposition pseudospectral method for nonlinear Fokker-Planck equations, Commun. Appl. Math. Comput. 1 (2019), pp. 231–252.
   Google Scholar
• T. Tang, The Hermite spectral method for Gauss-type function, SIAM J. Sci. Comput. 14 (1993), pp. 594–606.
   Web of Science ®Google Scholar
• T.J. Wang, Composite generalized Laguerre spectral method for nonlinear Fokker-Planck equations on the whole line, Math. Meth. Appl. Sci. 40 (2016), pp. 1462–1474.
   Web of Science ®Google Scholar
• T.J. Wang and G. Chai, A fully discrete pseudospectral method for the nonlinear Fokker-Planck equations on the whole line, Appl. Numer. Math. 174 (2022), pp. 17–33.
   Web of Science ®Google Scholar
• Z.Q. Wang and B.Y. Guo, Jacobi rational approximation and spectral method for differential equations of degenerate type, Math. Comp. 77 (2008), pp. 883–908.
   Web of Science ®Google Scholar
• T.J. Wang and B.Y. Guo, Composite generalized Laguerre-Legendre pseudospectral method for exterior problems, Commun. Comput. Phys. 5 (2009), pp. 350–375.
   Web of Science ®Google Scholar
• T.J. Wang and Y.J. Jiao, A fully discrete pseudospectral method for Fisher's equation on the whole line, Appl. Numer. Math. 120 (2017), pp. 243–256.
   Web of Science ®Google Scholar
• Z.Q. Wang and C.T. Sheng, An hp-spectral collocation method for nonlinear Volterra integral equations with vanishing variable delays, Math. Comp. 85 (2016), pp. 635–666.
   Web of Science ®Google Scholar
• Z.Q. Wang, Y.L. Guo, and L.J. Yi, An hp-version Legendre-Jacobi spectral collocation method for Volterra integral-differential equations with smooth and weak singular kernels, Math. Comp. 86 (2017), pp. 2285–2324.
   Web of Science ®Google Scholar
• H. Wu, H.P. Ma, and H.Y. Li, Optimal error estimates of the Chebyshev-Legendre spectral method for solving the generalized Burgers equation, SIAM J. Numer. Anal. 41 (2004), pp. 659–672.
   Web of Science ®Google Scholar
• S.S. Xie, G.X. Li, and S. Yi, A compact finite difference method for solving Burgers' equation, Int. J. Numer. Meth. Fluids 62 (2010), pp. 747–764.
   Web of Science ®Google Scholar
• X.M. Xing and Z.Q. Wang, Generalized Hermite spectral method and its applications to problems in unbounded domains, SIAM J. Numer. Anal. 48 (2010), pp. 1231–1253.
   Web of Science ®Google Scholar
• L.J. Yi and B.Q. Guo, An hp-version continuous Petrov-Galerkin finite element method for Volterra integral-differential equations with smooth and nonsmooth singular kernels, SIAM J. Numer. Anal. 53 (2015), pp. 2677–2704.
   Web of Science ®Google Scholar
• L. Zhang, L. Wang, and X. Ding, Exact finite difference scheme and nonstandard finite difference scheme for Burgers and Burgers-Fisher equations, J. Appl. Math. 122 (2014), pp. 1–12.
   Google Scholar