Existence and Uniqueness of Mild Solutions for the Damped Burgers Equation in Weighted Sobolev Spaces on the Half Line

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Mohammadreza Foroutan
Ali Ebadian

Abstract

This paper addresses an initial boundary value problem for the damped Burgers equation in weighted Sobolev spaces on half line. First, it introduces two normed spaces and present relations between them, which in turn enables us to analysis the existence and uniqueness of a local mild solution and of a global strong solution in these weighted spaces. The paper also studies the well-posedness of this equation in a semi-infinite interval.

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References

  1. R. Abazari, Application of Extended Tanh Function Method on KdV-Burgers Equation with Forcing Term, Rom. J. Phys. 59(1-2) (2014), 3-11.
  2. A. Biswas, S. Kumar, E.V. Krishnan, B. Ahmed, A. Strong, S. Johnson, A. Yildirim Topological Solitons and Other Solutions to Potential KdV Equation, Rom. J. Phys. 65(4) (2013), 1125-1137.
  3. J.M. Burgers, A mathematical model illustrating the theory of turbulence, Adv. Appl. Mech. 1 (1948), 171-199.
  4. J.D. Cole, On a puasilinear parabolic equations occurring in aerodynamics, Quart. Appl. Math. 9 (1951), 225-236.
  5. M. Dehghan, B.N. Saray and M. Lakestani, Mixed finite difference and Galerkin methods for solving Burgers equations using interpolating scaling functions, Math. Meth. Appl. Sci. 37 (6) (2014), 894-912.
  6. G. Ebadi, A. Mojaver, H. Triki, A. Yildirim, A. Biswas, Topological solitons and other solutions of the RosenauCKdV equation with power law nonlinearity, Rom. J. Phys. 58 (2013), 3-14.
  7. G. Ebadi, N. Yousefzadeh, H. Triki, A. Yildirim, A. Biswas, Envolope solitons, peridic waves and other solutions to BoussinesqCBurgers equstion, Rom. J. Phys. 64 (2012), 915-932.
  8. D. Goubet and J. Shen, On the dual Petrov-Galerkin formulation of the KdV equation on a finite interval, Adv. Diff. Equs. 12 (2007), 221-239.
  9. P. Grisvard, El liptic problems in nonsmooth domains, Volume 24 of monographs and studies in Mathematics, Pitman (Advanced Publishing Program), Boston, MA (1985).
  10. D. Idris, C.I. Aynur and S. Ali, Taylor Galerkin and Taylor collocation methods for the numerical solutions of Burgers equation using B-splines, Commun. Nonlinear Sci. Numer. Simul. 16:7 (2011), 2696-2708.
  11. I.E. Inan, D. Kaya and Y. Ugurlu, Auto Backlund transformation and similarity reductions for coupled Burgers equation, Appl. Math. Comput. 216 (2010), 2507-2511.
  12. K. Ismail and S. Ibrahim, An efficient computational method for the optimal control problem for the Burgers equation, Math. Comput. Model. 44 (2006), 973-982.
  13. M. Javidi, A numerical solution of Burgers equation based on modified extended BDF scheme, Int. Math. Forum 1,(2006), 1565-1570.
  14. N. Khanal, J. Wu and J.M. Yuan, The Kawahara equation in weighted Sobolev spaces, Nonlinearity 21 (2008), 1489-1505.
  15. A.H. Khater, R.S. Temsah and M.M. Hassan, A Chebyshev spectral collocation method for solving Burgers-type equation, J. Comput. Appl. Math. 222 (2008), 333-350.
  16. M. K. Kunisch and S. Volkwein, Control of the Burgers equation by a reduced-order approach using proper orthogonal decomposition, J. Optim. Theory Appl. 102 (1999), 345-371.
  17. Jerome L.V. Lewandowski, A marker method for the solution of the damped Burgers equation, Numer. Methods Partial Differ. 22(1) (2005), 48-68.
  18. J.L. Lions, Sur les problems aux limites du type derive oblique, Ann. Math. 64 (1956), 207-239.
  19. S. Lu and M. Li, Modified Legendre rational spectral method for thr Burgers equation on the half line, Int. J. Comput. Math. 85:6 (2008), 865-875.
  20. W. Malfliet, Approximate solution of the damped Burgers equation, J. Phys. A: Math. Gen. 26 (1993), L723-L728.
  21. R.C. Mittal and G. Arora, Numerical solution of the coupled viscous Burgers equation, Commun. Nonlinear Sci. Numer. Simul. 16 (2011), 1304-1313.
  22. A. Pazy, Semigroups of linear operators and applications to partial differential equations, New York: Springer, (1983).
  23. Y. Peng, W. Chen, A new similarity solution of the Burgers equation with linear damping Czech, J, Phys. 56 (2008) 317-428.
  24. Z. Sabeh, M. Shamsi and M. Dehghan, Distributed optimal control of the viscous Burgers equation via a Legendre pseudo- spectral approach, Math. Meth. Appl. Sci. 39(12) (2015),3350-3360
  25. H. Triki, A. Yildirim, T. Hayat, O.M. Aldossary, A. Biswas, Topological and non-topological solitons of a generalized derivative nonlinear Schrodingers equation with perturbation terms, Rom. J. Phys. 64 (2012), 672-684.
  26. F. Yilmaz and B. Karasozen, Solving optimal control problems for the unsteady Burgers equation in COMSOL multiphysics, J. Comput. Appl. Math. 235 (16) (2011), 4839-4850.
  27. N.Y. Fard, M.R. Foroutan, M. Eslami, M. Mirzazadeh, A. Biswas, Solitary waves and other solutions to Kadomtsev- Petviashvili equation with spatio-temporal dispersion, Rom. J. Phys. 60 (2015), 1337-1360.
  28. B. M. Vaganan, M. S. Kumaran, Kummer function solutions of damped Burgers equations with time-dependent viscosity by exact linearization, Nonlinear Anal. Real World Appl. 4 (2003), 723-741.