Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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The generalized quasilinearization method for parabolic integro-differential equations


Authors: A. S. Vatsala and Liwen Wang
Journal: Quart. Appl. Math. 59 (2001), 459-470
MSC: Primary 35K60; Secondary 35B05, 45K05
DOI: https://doi.org/10.1090/qam/1848528
MathSciNet review: MR1848528
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Abstract: In this paper we consider the nonlinear parabolic integro-differential equation with initial and boundary conditions. We develop the method of generalized quasilinearization to generate linear iterates that converge quadratically to the unique solution of the nonlinear parabolic integro-differential equation. For this purpose, we establish comparison results for the parabolic integro-differential equation. These comparison results are used to develop monotone sequences and to establish quadratic convergence.


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  • [1] Richard Bellman, Methods of nonliner analysis. Vol. II, Academic Press, New York-London, 1973. Mathematics in Science and Engineering, Vol. 61-II. MR 0381408
  • [2] Richard E. Bellman and Robert E. Kalaba, Quasilinearization and nonlinear boundary-value problems, Modern Analytic and Computional Methods in Science and Mathematics, Vol. 3, American Elsevier Publishing Co., Inc., New York, 1965. MR 0178571
  • [3] John R. Cannon and Yan Ping Lin, Smooth solutions for an integro-differential equation of parabolic type, Differential Integral Equations 2 (1989), no. 1, 111–121. MR 960018
  • [4] G. S. Ladde, V. Lakshmikantham, and A. S. Vatsala, Monotone iterative techniques for nonlinear differential equations, Monographs, Advanced Texts and Surveys in Pure and Applied Mathematics, vol. 27, Pitman (Advanced Publishing Program), Boston, MA; distributed by John Wiley & Sons, Inc., New York, 1985. MR 855240
  • [5] V. Lakshmikantham and S. Leela, Differential and Integral Inequalities, vol. VII, Academic Press, New York, 1968
  • [6] V. Lakshmikantham and M. Rama Mohana Rao, Theory of integro-differential equations, Stability and Control: Theory, Methods and Applications, vol. 1, Gordon and Breach Science Publishers, Lausanne, 1995. MR 1336142
  • [7] V. Lakshmikantham and A. S. Vatsala, Generalized quasilinearization for nonlinear problems, Mathematics and its Applications, vol. 440, Kluwer Academic Publishers, Dordrecht, 1998. MR 1640601
  • [8] C. V. Pao, Nonlinear parabolic and elliptic equations, Plenum Press, New York, 1992. MR 1212084
  • [9] S. G. Deo and C. McGloin Knoll, Further extension of the method of quasi-linearization to integro-differential equations, International Journal of Nonlinear Differential Equations: Theory, Methods, and Applications, Vol. 3, 1997, pp. 91-103
  • [10] A. S. Vatsala, Generalized quasilinearization and reaction diffusion equations, Nonlinear Times Digest 1 (1994), no. 2, 211–220. MR 1298578
  • [11] Donna Stutson and A. S. Vatsala, Quadratic and semi-quadratic convergence of IVP, Neural Parallel Sci. Comput. 3 (1995), no. 2, 235–248. MR 1345739

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Additional Information

DOI: https://doi.org/10.1090/qam/1848528
Article copyright: © Copyright 2001 American Mathematical Society


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