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
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
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.
- Richard Bellman, Methods of nonliner analysis. Vol. II, Academic Press, New York-London, 1973. Mathematics in Science and Engineering, Vol. 61-II. MR 0381408
- 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
- 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
- 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
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, vol. 1, Gordon and Breach Science Publishers, Lausanne, 1995. MR 1336142
- V. Lakshmikantham and A. S. Vatsala, Generalized quasilinearization for nonlinear problems, Mathematics and its Applications, vol. 440, Kluwer Academic Publishers, Dordrecht, 1998. MR 1640601
- C. V. Pao, Nonlinear parabolic and elliptic equations, Plenum Press, New York, 1992. MR 1212084
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 Digest 1 (1994), no. 2, 211–220. MR 1298578
- 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
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)
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC:
35K60,
35B05,
45K05
Retrieve articles in all journals
with MSC:
35K60,
35B05,
45K05
Additional Information
Article copyright:
© Copyright 2001
American Mathematical Society