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Original Articles

Legendre–Galerkin spectral approximation for a nonlocal elliptic Kirchhof-type problem

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Pages 1747-1758 | Received 22 Jan 2016, Accepted 06 Aug 2016, Published online: 13 Sep 2016

References

  • T. Abesadze, V. Odisharia, and J. Peradze, On the influence of a calculation error on the solution of the Kirchhoff nonlinear static equation, Seminar of I. Vekua Institute of Applied Mathematics, Reports, 2003, pp. 3–10.
  • C.O. Alves, F.J.S.A. Corrêa, and T.F. Ma, Positive solutions for a quasilinear elliptic equation of Kirchhoff type, Comput. Math. Appl. 49(1) (2005), pp. 85–93. doi: 10.1016/j.camwa.2005.01.008
  • C.O. Alves, F.J.S.A. Correa, and G.M. Figueiredo, On a class of nonlocal elliptic problems with critical growth, Differ. Equ. Appl. 2(3) (2010), pp. 409–417.
  • A. Arosio and S. Panizzi, On the well-posedness of the Kirchhoff string, Trans. Amer. Math. Soc. 348(1) (1996), pp. 305–330. doi: 10.1090/S0002-9947-96-01532-2
  • C. Canuto, M. Hussaini, A. Quarteroni, and T. Zang, Spectral Methods: Fundamentals in Single Domains, Springer, Berlin, 2006.
  • B. Cheng, New existence and multiplicity of nontrivial solutions for nonlocal elliptic Kirchhoff type problems, J. Math. Anal. Appl. 394(2) (2012), pp. 488–495. doi: 10.1016/j.jmaa.2012.04.025
  • B. Cheng and X. Wu, Existence results of positive solutions of Kirchhoff type problems, Nonlinear Anal: Theory, Methods Appl. 71(10) (2009), pp. 4883–4892. doi: 10.1016/j.na.2009.03.065
  • M. Chipot, V. Valente, and G. Vergara Caffarelli, Remarks on a nonlocal problem involving the Dirichlet energy, Rend. Sem. Mat. Univ. Padova 110(4) (2003), pp. 199–220.
  • P. Ciarlet and J. Lions, Handbook of Numerical Analysis, Vol. V: Techniques of Scientific Computing (Part 2), North-Holland, Amsterdam, 2000.
  • T. Gudi, Finite element method for a nonlocal problem of Kirchhoff type, SIAM J. Numer. Anal. 50(2) (2012), pp. 657–668. doi: 10.1137/110822931
  • B. Guo, Spectral Methods and Their Applications, World Scientific, Singapore, 1998.
  • G. Kirchhoff, Vorlesungen über mathematische Physik, Vol. 1, Teubner, Leipzig, 1883.
  • T.F. Ma, Remarks on an elliptic equation of Kirchhoff type, Nonlinear Anal: Theory, Methods Appl. 63(5) (2005), pp. e1967–e1977. doi: 10.1016/j.na.2005.03.021
  • J. Peradze, A numerical algorithm for the nonlinear Kirchhoff string equation, Numer. Math. 102(2) (2005), pp. 311–342. doi: 10.1007/s00211-005-0642-1
  • J. Peradze, An approximate algorithm for a Kirchhoff wave equation, SIAM J. Numer. Anal. 47(3) (2009), pp. 2243–2268. doi: 10.1137/070711876
  • K. Perera and Z. Zhang, Nontrivial solutions of Kirchhoff-type problems via the Yang index, J. Differ. Equ. 221(1) (2006), pp. 246–255. doi: 10.1016/j.jde.2005.03.006
  • H. Scarf, The approximation of fixed points of a continuous mapping, SIAM J. Appl. Math. 15(5) (1967), pp. 1328–1343. doi: 10.1137/0115116
  • B. Semper, Finite element methods for suspension bridge models, Comput. Math. Appl. 26(5) (1993), pp. 77–91. doi: 10.1016/0898-1221(93)90076-8
  • J. Shen, Efficient spectral-Galerkin method I. Direct solvers for the second- and fourth-order equations using Legendre polynomials, SIAM J. Sci. Comput. 15(6) (1994), pp. 1489–1505. doi: 10.1137/0915089
  • J. Shen, T. Tang, and L. Wang, Spectral Methods: Algorithms, Analysis and Applications, Springer, Berlin, 2011.
  • Y. Yang and J. Zhang, Positive and negative solutions of a class of nonlocal problems, Nonlinear Anal: Theory, Methods Appl. 73(1) (2010), pp. 25–30. doi: 10.1016/j.na.2010.02.008
  • E. Zeidler, Nonlinear Functional Analysis, Its Applications: Part 1, Springer, New York, 1986.
  • W. Zou and X. He, Multiplicity of solutions for a class of Kirchhoff type problems, Acta Math. Appl. Sin. English Series 26(3) (2010), pp. 387–394. doi: 10.1007/s10255-010-0005-2

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