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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R. Bellman, Methods of Nonlinear Analysis, Vol. II, Academic Press, New York, 1973
R. Bellman and R. Kalaba, Quasilinearization and Nonlinear Boundary Value Problems, American Elsevier, New York, 1965
J. Cannon and Y. P. Lin, Smooth solutions for an integro-differential equation of parabolic type, Differential and Integral Equations 2, 111–121 (1989)
G. S. Ladde, V. Lakshmikantham, and A. S. Vatsala, Monotone Iterative Techniques for Nonlinear Differential Equations, Pitman, Boston, 1985
V. Lakshmikantham and S. Leela, Differential and Integral Inequalities, vol. VII, Academic Press, New York, 1968
V. Lakshmikantham and M. Rama Mohana Rao, Theory of Integro-Differential Equations, Stability and Control: Theory, Methods and Applications, Volume 1, Gordon and Breach Science Publishers, Lausanne, Switzerland, 1995
V. Lakshmikantham and A. S. Vatsala, Generalized quasilinearization for nonlinear problems, Mathematics and its Applications 440, Kluwer Academic Publishers, Dordrecht, 1998
C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992
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
A. S. Vatsala, Generalized quasilinearization and reaction diffusion equations, Nonlinear Times and Digest 1, 211–220 (1994)
D. Stutson and A. S. Vatsala, Quadratic and semi-quadratic convergence of IVP, Neural, Parallel and Scientific Computations 3, 235–248 (1995)
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© Copyright 2001
American Mathematical Society
