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.

中文翻译：

---

一维非线性反应扩散方程的迭代解析逼近

该论文涉及一类具有耗散参数的非线性反应-扩散方程。从数学的角度来看，这个问题非常令人不安。已知这些问题的解决方案表现出多尺度特征。存在溶液具有陡峭梯度的狭窄区域。为了逼近多尺度解，我们提出并分析了一种基于拉格朗日乘子技术的迭代分析方法。我们通过迭代获得非线性边值问题的封闭形式解析近似。在一般情况下，使用变分理论和 Liouville-Green 变换可以最优地获得拉格朗日乘数。这篇论文的想法是克服与数值方法相关的众所周知的困难。考虑了两个测试示例，并进行了严格的对比分析。此外，我们将所提出的方法与文献中的其他方法进行了比较。