# 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 particular form, $$f(u)=u^3$$, the problem has a unique positive solution that satisfies an “isoperimetric” condition.

We study the bifurcation of this solution with respect to the length of the interval, simply by estimating its closed form using a generalized mean value theorem for integrals. It turns out that solution bifurcates 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 introducing 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.

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##### Publisher Name

Babes-Bolyai University

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0252-1938

2065-961x

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