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Research Article

An inverse problem for the damped generalized regularized long wave equation

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Pages 1395-1427 | Received 11 Jul 2020, Accepted 30 Aug 2021, Published online: 20 Sep 2021

References

  • T. Achouri, N. Khiari, and K. Omrani, On the convergence of difference schemes for the Benjamin–Bona–Mahony (BBM) equation, Appl. Math. Comput. 182 (2006), pp. 999–1005.
  • T. Achouri, M. Ayadi, and K. Omrani, A fully Galerkin method for the damped generalized regularized long-wave (DGRLW) equation, Numer. Methods Partial Differ. Equ. 25 (2009), pp. 668–684.
  • A.S. Alshomrani, S. Pandit, A.K. Alzahrani, M.S. Alghamdi, and R. Jiwari, A numerical algorithm based on modified cubic trigonometric B-spline functions for computational modelling of hyperbolic type wave equations, Eng. Comput. 34 (2017), pp. 1257–1276.
  • 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. 36 (1981), pp. 53–63.
  • T.B. Benjamin, J.L. Bona, and J.J. Mahony, Model equations for long waves in nonlinear dispersive systems, Phil. Trans. Royal Soc. Lond. A 272 (1972), pp. 47–78.
  • J.L. Bona, W.G. Pritchard, and L.R. Scott, An evaluation of a model equation for water waves, Philos. Trans. Royal Soc. Lond. Ser. A 302 (1981), pp. 457–510.
  • J.M.G. Cabeza, J.A.M. Garcia, and A.C. Rodriguez, A sequential algorithm of inverse heat conduction problems using singular value decomposition, Int. J. Therm. Sci. 44 (2005), pp. 235–244.
  • J. Caldwell, P. Wanless, and A.E. Cook, A finite element approach to Burgers' equation, Appl. Math. Model. 5 (1981), pp. 189–193.
  • İ. Çelik, Haar wavelet approximation for magnetohydrodynamic flow equations, Appl. Math. Model. 37 (2013), pp. 3894–3902.
  • D.D. Demir, N. Bildik, A. Konuralp, and A. Demir, The numerical solutions for the damped generalized regularized longwave equation by variational method, World Appl. Sci. J. 13 (2011), pp. 1308–1317.
  • D.B. Dix, The dissipation of nonlinear dispersive waves: the case of asymptotically weak nonlinearity, Comm. Partial Differ. Equ. 17 (1992), pp. 1665–1693.
  • D.B. Dix, Large-time behaviour of solutions of Burgers' equation, Proc. Roy. Soc. Edinburgh Sect. A132 (2002), pp. 843–878.
  • L. Elden, A note on the computation of the generalized cross-validation function for ill-conditioned least squares problems, BIT 24 (1984), pp. 467–472.
  • R.E. Ewing, Time-stepping Galerkin methods for nonlinear Sobolev partial differential equation, IAM J. Numer. Anal. 15 (1978), pp. 1125–1150.
  • G.H. Golub, M. Heath, and G. Wahba, Generalized cross-validation as a method for choosing a goodridge parameter, Technometrics 21 (1979), pp. 215–223.
  • C.A. Hall, On error bounds for spline interpolation, J. Approx. Theory 1 (1968), pp. 209–218.
  • C.A. Hall and W.W. Meyer, Optimal error bounds for cubic spline interpolation, J. Approx. Theory16 (1976), pp. 105–122.
  • P.C. Hansen, Analysis of discrete ill-posed problems by means of the L-curve, SIAM 34 (1992), pp. 561–580.
  • R. Jiwari, Haar wavelet quasilinearization approach for numerical simulation of Burgers' equation, Comput. Phys. Commun. 183 (2012), pp. 2413–2423.
  • R. Jiwari, A hybrid numerical scheme for the numerical solution of the Burgers' equation, Comput. Phys. Commun. 188 (2015), pp. 59–67.
  • R. Jiwari, Lagrange interpolation and modified cubic B-spline differential quadrature methods for solving hyperbolic partial differential equations with Dirichlet and Neumann boundary conditions, Comput. Phys. Commun. 193 (2015), pp. 55–65.
  • R. Jiwari, S. Pandit, and M.E. Koksal, A class of numerical algorithms based on cubic trigonometric B-spline functions for numerical simulation of nonlinear parabolic problems, Comput. Appl. Math. 38 (2019), pp. 1–18. https://doi.org/https://doi.org/10.1007/s40314-019-0918-1.
  • T. Kadri, N. Khiari, F. Abidi, and K. Omrani, Methods for the numerical solution of the Benjamin–Bona–Mahony–Burgers equation, Numer. Methods Partial Differ. Equ. 24(6) (2008), pp. 1501–1516.
  • C.L. Lawson and R.J. Hanson, Solving Least Squares Problems, SIAM, Philadelphia, PA, 1995.
  • Ü. Lepik, Numerical solution of differential equations using Haar wavelets, Math. Comput. Simul.68 (2005), pp. 127–143.
  • M. Limin and W. Zongmin, Radial basis functions method for parabolic inverse problem, Int. J. Cumput. Math. 88 (2011), pp. 384–395.
  • V. Marinca and N. Herişanu, Application of optimal homotopy asymptotic method for solving nonlinear equations arising in heat transfer, Int. Commun. Heat Mass Transfer 35 (2008), pp. 710–715.
  • V. Marinca, N. Herişanu, and I. Nemes, Optimal homotopy asymptotic method with application to thin film ow, Cent. Europ. J. Phys. 6 (2008), pp. 648–653.
  • V. Marinca, N. Herişanu, C. Bota, and B. Marinca, An optimal homotopy asymptotic method applied to the steady ow of a fourth-grade uid past a porous plate, Appl. Math. Lett. 22 (2009), pp. 245–251.
  • L. Martin, L. Elliott, P.J. Heggs, D.B. Ingham, D. Lesnic, and X. Wen, Dual reciprocity boundary element method solution of the Cauchy problem for Helmholtz-type equations with variable coefficients, J. Sound Vib. 297 (2006), pp. 89–105.
  • K. Omrani, The convergence of the fully discrete Galerkin approximations for the Benjamin–Bona–Mahony (BBM) equation, Appl. Math. Comput. 180 (2006), pp. 614–621.
  • K. Omrani and M. Ayadi, Finite difference discretization of the Benjamin–Bona–Mahony–Burgers (BBMB) equation, Numer. Methods Partial Differ. Equ. 24 (2008), pp. 239–248.
  • Ö. Oru, F. Bulut, and A. Esen, A Haar wavelet-finite difference hybrid method for the numerical solution of the modified Burgers' equation, J. Math. Chem. 53 (2015), pp. 1592–1607.
  • Ö. Oru, F. Bulut, and A. Esen, Numerical solutions of regularized long wave equation by Haar Wavelet method, Mediterr. J. Math. 13 (2016), pp. 3235–3253.
  • Ö. Oru, A. Esen, and F. Bulut, A Strang splitting approach combined with Chebyshev wavelets to solve the regularized long-wave equation numerically, Mediterr. J. Math. 17 (2020), pp. 140. https://doi.org/https://doi.org/10.1007/s00009-020-01572-w.
  • E. Ott and R.N. Sudan, Nonlinear theory of ion acoustic waves with Landau damping, Phys. Fluids12 (1969), pp. 2388–2394.
  • S. Pandit, R. Jiwari, K. Bedi, and M.E. Koksal, Haar wavelets operational matrix based algorithm for computational modelling of hyperbolic type wave equations, Eng. Comput. 34 (2017), pp. 793–814.
  • M.A. Ramadan, T.S. El-Danaf, and F.E.I. Abd Alaal, A numerical solution of the Burgers' equation using septic B-splines, Chaos Solitons Fractals 26 (2005), pp. 795–804.
  • J. Rashidinia, M. Ghasemi, and R. Jalilian, A collocation method for the solution of nonlinear onedimensional parabolic equations, Math. Sci. 4 (2010), pp. 87–104.
  • S.G. Rubin and R.A. Graves, A cubic spline approximation for problems in fluid mechanics, Tech. Rep. National Aeronautics and Space Administration, Washington, 1975.
  • R.S. Saha Ray, On Haar wavelet operational matrix of general order and its application for the numerical solution of fractional Bagley Torvik equation, Appl. Math. Comput. 218 (2012), pp. 5239–5248.
  • A.N. Tikhonov and V.Y. Arsenin, On the Solution of Ill-Posed Problems, Wiley, New York, 1977.
  • S.K. Turitsyn, J.J. Rasmussen, and M.A. Raadu, Stability of Weak Double Layers, Royal Institute of Technology, Stockholm, 1991. TRITA-EPP-91-01.
  • S.A. Yousef, Z. Barikbin, and M. Behroozifar, Bernstein Ritz-Galerkin method for solving the damped generalized regularized long-wave (DGRLW) equation, Int. J. Nonlinear Sci. 9 (2010), pp. 151–158.
  • G. Wahba, Spline models for observational data, CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 59, SIAM, Philadelphia, 1990.
  • L. Wahlbin, Error estimates for a Galerkin method for a class of model equations for long waves, Numer. Math. 23 (1975), pp. 289–303.

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