Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

THE METHOD OF QUASILINEARIZATION AND A THREE-POINT BOUNDARY VALUE PROBLEM

  • Eloe, Paul W. (Department of Mathematics, University of Dayton) ;
  • Gao, Yang (Department of Mathematics, University of Dayton)
  • Published : 2002.03.01
Download PDF
⟨ Previous Next ⟩

Abstract

The method of quasilinearization generates a monotone iteration scheme whose iterates converge quadratically to a unique solution of the problem at hand. In this paper, we apply the method to two families of three-point boundary value problems for second order ordinary differential equations: Linear boundary conditions and nonlinear boundary conditions are addressed independently. For linear boundary conditions, an appropriate Green\`s function is constructed. Fer nonlinear boundary conditions, we show that these nonlinearities can be addressed similarly to the nonlinearities in the differential equation.

Keywords

References

  1. Methods of Nonlinear Analysis R. Bellman
  2. Quasilinearization and Nonlinear Boundary Value Problems R. Bellman;R. Kalba
  3. Appl. Math. Comput. v.87 Rapid convergence of the iterative technique for first order initial value problems A. Cabada;J. Nieto https://doi.org/10.1016/S0096-3003(96)00285-8
  4. Int. J. Math. Math. Sci. v.21 Rapid convergence of approximate solutions first order nonlinear boundary value problems A. Cabada;J. Nieto;S. Heikkila https://doi.org/10.1155/S0161171298000714
  5. Dynam. Contin. Discrete Impuls. Systems v.4 A note on rapid convergence of approximate solutions for an ordinary Dirichlet problem A. Cabada;J. Nieto;R. Pita-da-Viega
  6. Commun. Appl. Anal. v.2 Monotone and quadratic convergence of approximate solutions of ordinary differential equations with impulse V. Doddaballapur;P. W. Eloe
  7. Electron. J. Differ. Equ., Conf. 01 Quadratic convergence of approximate solutions of two-point boundary value problems with impulse V. Doddaballapur;P. W. Eloe;Y. Zhang
  8. Nonlinear Anal. v.33 A quadratic monotone iteration scheme for two-point boundary value problems for ordinary differential equations P. W. Eloe;Y. Zhang, https://doi.org/10.1016/S0362-546X(97)00633-0
  9. J. Math. Anal. Appl. v.168 Solvability of a three-point boundary value problem for a second order ordinary differential equation C. P. Gupta https://doi.org/10.1016/0022-247X(92)90179-H
  10. Nonlinear Anal. v.24 A second order m-point boundary value problem at resonance https://doi.org/10.1016/0362-546X(94)00204-U
  11. J. Inequal. Appl. v.5 A priori estimates for the existence of a solution for a multi-point boundary value problem C. P. Gupta;S. Trofimchuk https://doi.org/10.1155/S1025583400000187
  12. Studies in Ordinary Differential Equations, MAA Studies in Mathematics, Vol. 14 Mathematical Association of America Boundary value problems for ordinary differential equations L. Jackson;J. K. Hale (Ed.),
  13. J. Math. Anal. Appl. v.101 In certain boundary value problems for second order linear ordinary differential equations with singularities I. T. Kiguradze;A. G. Lomtatidze https://doi.org/10.1016/0022-247X(84)90107-0
  14. Generalized Quasilinearization for Nonlinear Problems V. Lakschmikantham;A. S. Vatsala
  15. Tbiliss. Gos. Univ. Inst. Prikl. Mat. Trudy v.22 On the problem of the solvability of singular boundary value problems for second order ordinary di?erential equations A. G. Lomtatidze
  16. J. Math. Anal. Appl. v.193 On a nonlocal boundary value problem for second order ordinary differential equations https://doi.org/10.1006/jmaa.1995.1273
  17. Nonlinear Anal. v.36 Generalized quasilinearization method for second order boundary value problem R. N. Mohapatra;K. Vajravelu;Y. Yin https://doi.org/10.1016/S0362-546X(98)00131-X
  18. Proc. Amer. Math. Soc. v.125 Generalized quasilinearization method for a second order differential equation with Dirichlet boundary conditions J. Nieto https://doi.org/10.1090/S0002-9939-97-03976-2
  19. Applicable Anal. v.58 An extension of the method of generalized quasilinearization for second order boundary value problems N. Shahzad;A. S. Vatsala https://doi.org/10.1080/00036819508840363

Cited by

  1. Positive Solutions of Three-point Nonlinear Discrete Second Order Boundary Value Problem vol.10, pp.2, 2004, https://doi.org/10.1080/1023619031000114323
  2. Approximation of Solutions for Second-Order m-Point Nonlocal Boundary Value Problems via the Method of Generalized Quasilinearization vol.2011, 2011, https://doi.org/10.1155/2011/929061
  3. The generalized quasilinearization method and three point boundary value problems on time scales vol.18, pp.5, 2005, https://doi.org/10.1016/j.aml.2004.06.022
  4. Numerov's Method for a Class of Nonlinear Multipoint Boundary Value Problems vol.2012, 2012, https://doi.org/10.1155/2012/316852
  5. A quasilinearization method for second-order four-point boundary value problems vol.202, pp.1, 2008, https://doi.org/10.1016/j.amc.2008.01.026
  6. Monotonic Positive Solutions of Nonlocal Boundary Value Problems for a Second-Order Functional Differential Equation vol.2012, 2012, https://doi.org/10.1155/2012/489353
  7. Eigenvalues of a general class of boundary value problem with derivative-dependent nonlinearity vol.259, 2015, https://doi.org/10.1016/j.amc.2015.02.087
  8. Existence and approximation of solutions of second-order nonlinear four point boundary value problems vol.63, pp.8, 2005, https://doi.org/10.1016/j.na.2005.05.030
  9. Generalized approximations and rapid convergence of solutions of m-point boundary value problems vol.188, pp.2, 2007, https://doi.org/10.1016/j.amc.2006.11.138
  10. Existence and approximation of solutions of forced duffing type integro-differential equations with three-point nonlinear boundary conditions vol.11, pp.4, 2010, https://doi.org/10.1016/j.nonrwa.2009.10.014
  11. Approximations and rapid convergence of solutions of nonlinear three point boundary value problems vol.186, pp.2, 2007, https://doi.org/10.1016/j.amc.2006.08.045
  12. The generalized quasilinearization method for second-order three-point boundary value problems vol.68, pp.9, 2008, https://doi.org/10.1016/j.na.2007.02.025
  13. On nonlocal three-point boundary value problems of Duffing equation with mixed nonlinear forcing terms vol.2011, pp.1, 2011, https://doi.org/10.1186/1687-2770-2011-47