Hostname: page-component-76fb5796d-wq484 Total loading time: 0 Render date: 2024-04-28T07:46:46.968Z Has data issue: false hasContentIssue false
  • English
  • Français

General iterative methods for nonlinear boundary value problems

Published online by Cambridge University Press:  17 February 2009

Radha Shridharan  and
Ravi P. Agarwal
Radha Shridharan
Affiliation:
BLK 112, Tampines Street 11, # 08–195, Singapore1852.
Ravi P. Agarwal
Affiliation:
Department of Mathematics, National University of Singapore, Singapore 0511 for correspondence.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper we shall develop existence-uniqueness as well as constructive theory for the solutions of systems of nonlinear boundary value problems when only approximations of the fundamental matrix of the associated homogeneous linear differential systems are known. To make the analysis widely applicable, all the results are proved component-wise. An illustration which dwells upon the sharpness as well as the importance of the obtained results is also presented.

Type
Research Article
Information
The ANZIAM Journal , Volume 37 , Issue 1 , July 1995 , pp. 58 - 85
Copyright
Copyright © Australian Mathematical Society 1995

References

[1]Agarwal, R. P., “Contraction and approximate contraction with an application to multi-point boundary value problems”, J. Comp. Appl. Math. 9 (1983) 315–325.CrossRefGoogle Scholar
[2]Agarwal, R. P., “Component-wise convergence of iterative methods for non-linear Volterra integro- differential systems with nonlinear boundary conditions”, Math. Phyl. Sci. 18 (1984) 291–322.Google Scholar
[3]Agarwal, R. P., “On Urabe's application of Newton's method to nonlinear boundary value problems”, Arch. Math. (Bmö) 20 (1984) 113–124.Google Scholar
[4]Agarwal, R. P., “Computational methods for discrete boundary value problems”, Appl. Math. Comp. 18 (1986) 15–41.CrossRefGoogle Scholar
[5]Agarwal, R. P., “Component-wise convergence of quasilinear method for nonlinear boundary value problems”, Hiroshima Math. J. 22 (1992) 525–541.CrossRefGoogle Scholar
[6]Agarwal, R. P., “Computational methods for discrete boundary value problems II”, J. Math. Anal. Appl. 166 (1992) 540–562.CrossRefGoogle Scholar
[7]Agarwal, R. P. and Vòsmanskỳ, J., “Necessary and sufficient conditions for the convergence of approximate Picard's iterates for nonlinear boundary value problems”, Arch. Math. (Brnö) 21 (1985) 171–176.Google Scholar
[8]Bernfeld, S. R. and Lakshmikantham, V., An introduction to nonlinear boundary value problems (Academic Press, New York, 1974).Google Scholar
[9]Fujii, M., “An a posteriori error estimation of the numerical solution by step-by-step methods for systems of ordinary differential equations”, Bull. Fukoka Univ. Ed. III 23 (1973) 35–44.Google Scholar
[10]Fujii, M. and Hayashi, Y., “Numerical solutions to problems of the least squares type for ordinary differential equations”, Hiroshima Math. J. 13 (1983) 477–499.CrossRefGoogle Scholar
[11]Hayashi, Y., “On a posteriori error estimation in the numerical solution of systems of ordinary differential equations”, Hiroshima Math. J. 9 (1979) 201–243.Google Scholar
[12]Kurpel’, N. S., Projection-iterative methods for solutions of operator equations, Translations of Math. Monographs (Amer. Math. Soc., Providence, Rhode Island, 1976).Google Scholar
[13]Mitsui, T., “The initial-value adjusting method for problems of the least square type of ordinary differential equations”, Publ. RIMS — Kyoto Univ. 16 (1980) 785–810.Google Scholar
[14]Mitsui, T., “Iterative approximation methods for non-linear boundary value problems of ODE” Lecture Notes in Num. Appl. Anal. (The Newton Method and Related Topics) 3 (1981) 105–125.Google Scholar
[15]Ojika, T., “On quadratic convergence of the initial estimates and iterative methods for nonlinear multipoint boundary value problems”, J. Math. Anal. Appl. 73 (1980) 192–203.Google Scholar
[16]Ojika, T. and Kasue, T., “Initial-value adjusting method for the solution of nonlinear multipoint boundary-value problems”, J. Math. Anal. Appl. 69 (1979) 359–371.Google Scholar
[17]Ojika, T. and Welsh, W., “A numerical method for the solution of multi-point problems for ordinary differential equations with integral constraints”, J. Math. Anal. Appl. 72 (1979) 500–511.CrossRefGoogle Scholar
[18]Schröder, J., Operator inequalities (Academic Press, New York, 1980).Google Scholar
[19]Shintani, H. and Hayashi, Y., “A posteriori error estimates and iterative methods in the numerical solution of systems of ordinary differential equations”, Hiroshima Math. J. 8 (1978) 101–121.Google Scholar
[20]Shridharan, Radha and Agarwal, R. P., “Stationary and nonstationary iterative methods for nonlinear boundary value problems”, Math. Comp. Modelling 18 (1993) 43–62.Google Scholar
[21]Teo, K. L., Goh, C. J. and Wong, K. H., A unified computational approach to optimal control problems (Longman Scientific and Technical, Essex, England, 1991).Google Scholar
[22]Urabe, M., “An existence theorem for multi-point boundary value problems”, Funkcialaj Ekvacioj 9 (1966) 43–60.Google Scholar
[23]Urabe, M., “The Newton method and its application to boundary value problems with nonlinear boundary conditions”, in Proc. US - Japan Seminar on Differential and Functional Equations (Benjamin, 1967) 383–410.Google Scholar
[24]Urabe, M., “On the Newton method to solve problems of the least square type for ordinary differential equations”, Memoirs of the Faculty of Sci. Kyushu Univ. Ser. A 29 (1975) 173–183.Google Scholar
[25]Yamamoto, T., “An existence theorem of solution to boundary value problems and its applications to error estimates”, Math. Japonica 27 (1982) 301–318.Google Scholar