On the bifurcation and variational approximation of the positive solution of a nonlinear reaction-diffusion problem

Abstract

We consider a nonlinear, second-order, two-point boundary value problem that models some reaction-diffusion processes. When the reaction term has a particula form, f(u)=u³, the problem has a unique positive solution that satisfies an “isoperimetric” condition. We study the bifurcation of this solution with respect to the lenght of the interval, simply by estimating its closed form using a generalized mean value theorem for integrals. It turns out that solution bifurcation from infinity. In order to obtain directly accurate numerical approximations to this positive solution, we characterize it by a variational problem involving a conditional extremum. We built up a functional to be minimized introcuing the “isoperimetric” condition as a restriction by Lagrange’s multiplier method. We then derive a necessary condition of extremum to be satisfied by the weak approximation and carry out some numerical results. They are in good agreement with those previously obtained by indirect methods.

Authors

C.I. Gheorghiu
Tiberiu Popoviciu Institute of Numerical Analysis

D. Trif
Babes-Bolyai University

Keywords

nonlinear reaction-diffusion; positive solution; bifurcation from infinity; conditional extremum; f.e.m.

References

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Paper coordinates

C.I. Gheorghiu, D. Trif, On the bifurcation and variational approximation of the positive solution of a nonlinear reaction-diffusion problem, Studia Univ. Babeş-Bolyai Math., XLV (2000) no.3, 29-37.

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Babes-Bolyai University

Print ISSN

0252-1938

Online ISSN

2065-961x

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References

[1] Elsgolts, L., Differential equations and the calculus of varations,  Mir. Publishers, Moscow,, 1980.

[2] Gheorghiu, C.I., Solution to problem 97-8, by Ph Korman, “Average temperature in a reaction-diffusion problem”, SIAM Review, 40, 2, pp. 382-385, 1998.

[3] Gheorghiu, C.I., Trif, D.,
 The numerical approximation to positive solution for some  reaction-diffusion problems,  Pu.M.A., to appear.

[4] Moore, R. A., Nehari, Z., Nonoscillation Theorems for a class of nonlinear differential equations,  Trans. Amer. Math. Soc., 93, pp. 30-52, 1959.

[5] Schwind, W. J., Ji. J., Koditschek, D. E.,
 A physically motivated further note on the mean value theorem for integrals,  Amer. Math. Monthly, 126, pp. 559-564, 1999.

[6] Wollkind, D. J., Monoranjan, V. S., Zhang, L.,
Weakly nonlinear stability analysis of prototype reaction-diffusion model equations, SIAM Review, 36, 2, pp. 176-214, 1994.

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