Iterative analytic approximation to one-dimensional nonlinear reaction-diffusion equations

The paper is concerned with a class of nonlinear reaction-diffusion equations with a dissipating parameter. The problem is singularly perturbed from a mathematical perspective. Solutions of these problems are known to exhibit multiscale character. There are narrow regions in which the solution has a steep gradient. To approximate the multiscale solution, we present and analyze an iterative analytic method based on a Lagrange multiplier technique. We obtain closed-form analytic approximation to nonlinear boundary value problems through iteration. The Lagrange multiplier is obtained optimally, in a general setting, using variational theory and Liouville-Green transforms. The idea of the paper is to overcome the well-known difficulties associated with numerical methods. Two test examples are taken into account, and rigorous comparative analysis is presented. Moreover, we compare the proposed method with others found in the literature.