SIAM Journal on Applied Mathematics, Volume 81, Issue 1, Page 208-232, January 2021.
A nonlinear reaction-diffusion partial differential equation occurring in models of transient controlled-potential experiments in electroanalytical chemistry is investigated analytically and numerically, with a view to determining a relationship between the concentration of a chemical species and its flux at a reacting electrode. It is shown that a previously known relation that holds for the steady-state case can be used as the first term in a singular perturbation expansion for the time-dependent case. However, in trying to determine the second term, so as to extend the range of validity of the solution, it is found that a phenomenon akin to switchbacking occurs, with the asymptotic details being strongly dependent on the reaction order; this appears to be a consequence of the spatial algebraic decay of the leading-order solution far from the electrode. Comparison of asymptotic results with numerical solutions obtained using finite element methods indicates a relation involving the homogeoneous reaction order for which the two-term asymptotic approximation would work best for all time. Links to problems that involve algebraically decaying boundary layers are briefly discussed.

中文翻译：

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电分析化学的非线性瞬态反应扩散问题

SIAM应用数学杂志，第81卷，第1期，第208-232页，2021年1月。
为了确定化学物质的浓度与其在反应电极上的通量之间的关系，对在分析化学中的瞬态受控电位实验模型中出现的非线性反应扩散偏微分方程进行了分析和数值研究。已经表明，对于稳态情况，先前已知的关系可以用作时间相关情况的奇异摄动展开中的第一项。然而，在试图确定第二项以扩展解的有效性范围时，发现出现了类似于折返的现象，其渐近细节在很大程度上取决于反应顺序。这似乎是远离电极的前导解的空间代数衰减的结果。渐近结果与使用有限元方法获得的数值解的比较表明，一个涉及同质反应阶数的关系，对于该关系，二项渐近逼近在所有时间内都最有效。简要讨论了涉及代数衰减边界层的问题的链接。