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.
Tiberiu Popoviciu Institute of Numerical Analysis
nonlinear reaction-diffusion; positive solution; bifurcation from infinity; conditional extremum; f.e.m.
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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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