# 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

See the expanding block below.

## 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.

## PDF

##### Publisher Name

Babes-Bolyai University

0252-1938

2065-961x

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## References

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The numerical approximation to positive solution for some  reaction-diffusion problems,  Pu.M.A., to appear.

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