Direct and indirect approximations to positive solution for a nonlinear reaction-diffusion problem. I. Direct (variational)

Abstract

We consider a nonlinear, second-order, two-point boundary value problem that models some reaction-diffusion precesses. When the reaction term has a particular form, \(f(u)=u^{3}\), the problem has a unique positive solution that satisfies a conserved integral condition. We study the bifurcation of this solution with respect to the length of the interval and it turns out that solution bifurcates from infinity. In the first part, we obtain the numerical approximation to the positive solution by direct variational methods, while in the second part we consider indirect numerical methods. In order to obtain directly accurate numerical approximations to this positive solution, we characterize it by a variational problem involving a conditional extremum. Then we carry out some numerical experiments by usual finite elements method.

Keywords

• nonlinear reaction-diffusion
• positive solution
• conserved integral
• bifurcation
• variational formulation
• Lagrange multiplier