Summary
A new method is proposed for the inclusion of the critical parameter λ* of some convex operator equationu=λTu (appearing e.g. in thermal explosion theory). It is based on the fact that for a fixed λ Newton's method starting with a suitable subsolution is not monotonically if and only if λ>λ*. Several numerical examples arising from nonlinear boundary value problems illustrate the efficiency of the method.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Amann, H.: Fixed Point Equations and Nonlinear Eigenvalue Problems in Ordered Banach Spaces, SIAM Review,18, 620–709 (1976)
Amann, H.: Supersolutions, Monotone Iterations and Stability. J. Differential Equations21, 363–377 (1976)
Aris, R.: The Mathematical Theory of Diffusion and Reaction in Permeable Catalysts. Vols. I, II, Oxford: Clarendon Press 1975
Bandle, C.: Existence Theorems, Qualitative Results and a Priori Bounds for a Class of Nonlinear Dirichlet Problems. Arch. Rat. Mech. Anal.58, 219–238 (1975)
Bandle, C., Marcus, M.: Comparison Theorems for a Class of Nonlinear Dirichlet Problems. J. Differential Equations26, 321–334 (1977)
Bohl, E.: Monotonie: Lösbarkeit und Numerik bei Operatorgleichungen. Berlin-Heidelberg-New York: Springer 1974
Collatz, L.: Funktionalanalysis und Numerische Mathematik. Berlin-Göttingen-Heidelberg: Springer 1964
Collatz, L.: The Numerical Treatment of Differential Equations. 3rd ed., Berlin-Heidelberg-New York: Springer 1966
Frank-Kamenetskii, D.A.: Diffusion and Heat Transfer in Chemical Kinetics. New York-London: Plenum 1969
Joseph, D.D.: Variable Viscosity Effects on the Flow and Stability of Flow in Channels and Pipes. Phys. Fluids7, 1761–1771 (1964)
Keller, H.B.: Approximation Methods for Nonlinear Problems with Application to Two-Point Boundary Value Problems. Math. Comput.29, 464–474 (1975)
Keller, H.B., Cohen, D.S.: Some Positone Problems Suggested by Nonlinear Heat Generation. J. Math. Mech.16, 1361–1376 (1967)
Laetsch, T.: Asymptotic Branch Points and Multiple Positive Solution of Nonlinear Integral Equations. SIAM J. Math. Anal.6, 178–191 (1975)
Mooney, J.W., Roach, G.F.: Iterative Bounds for the Stable Solutions of Convex Nonlinear Boundary Value Problems. Proc. Roy. Soc. Edinburgh, 76A, 81–94 (1976)
Simpson, R.B.: Existence and Error Estimates for Solution of a Discrete Analog of Nonlinear Eigenvalue Problems. Math. Comput.26, 359–375 (1972)
Wake, G.C.: An Improved Bound for the Critical Explosion Condition for an Exothermic Reaction in an Arbitrary Shape. Combustion and Flame17, 171–174 (1971)
Wake, G.C., Rayner, M.E.: Variational Methods for Nonlinear Eigenvalue Problems Associated with Thermal Ignation. J. Differential Equations13, 247–256 (1973)
Weiss, R.: On the Approximation of Fixed Points of Nonlinear Compact Operators. SIAM J. Num. Anal.11, 550–553 (1974)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Mooney, J.W., Voss, H. & Werner, B. The dependence of critical parameter bounds on the monotonicity of a Newton sequence. Numer. Math. 33, 291–301 (1979). https://doi.org/10.1007/BF01398645
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01398645