Regions of existence and uniqueness for singular nonlinear diffusion problems

IF 1.7 3区化学 Q3 CHEMISTRY, MULTIDISCIPLINARY

Journal of Mathematical Chemistry Pub Date : 2024-12-14 DOI:10.1007/s10910-024-01700-x

Shih-Hsiang Chang

引用次数: 0

Abstract

This paper presents a novel approach for constructing the lower and upper boundaries of closed regions where solutions to the singular nonlinear diffusion problems

$$\begin{aligned} \begin{aligned} y''(x)+ \frac{m}{x}y'(x)= f(x,y(x)), \quad x \in (0,1], \quad m \ge 0 , \\ y'(0) = 0, \quad Ay(1)+By'(1) = C, \quad A>0, B \ge 0, C \ge 0 , \end{aligned} \end{aligned}$$

exist. This existence result is proved using the method of lower and upper solutions with monotone iterative technique under the restriction that f(x, y) is continuous in $x \in [0,1]$ and non-increasing in y in such regions. Additional uniqueness criteria is also established. The approach is illustrated on four singular nonlinear diffusion problems including some real life applications.

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本文提出了一种新方法，用于构建奇异非线性扩散问题$$\begin{aligned}解所在封闭区域的下边界和上边界。\y''(x)+frac{m}{x}y'(x)= f(x,y(x)), quad x in (0,1], quad m 0 , y'(0) = 0, quad Ay(1)+By'(1) = C, quad A>0, B 0, C 0 , end{aligned}.\end{aligned}$$存在。在 f(x, y) 在 $x \in [0,1]$ 中连续且在这些区域中 y 非递增的限制条件下，使用单调迭代技术的上下限解法证明了这一存在性结果。此外，还建立了额外的唯一性标准。该方法在四个奇异非线性扩散问题上进行了说明，包括一些实际应用。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Journal of Mathematical Chemistry 化学-化学综合

CiteScore

3.70

自引率

17.60%

发文量

105

审稿时长

6 months

期刊介绍： The Journal of Mathematical Chemistry (JOMC) publishes original, chemically important mathematical results which use non-routine mathematical methodologies often unfamiliar to the usual audience of mainstream experimental and theoretical chemistry journals. Furthermore JOMC publishes papers on novel applications of more familiar mathematical techniques and analyses of chemical problems which indicate the need for new mathematical approaches. Mathematical chemistry is a truly interdisciplinary subject, a field of rapidly growing importance. As chemistry becomes more and more amenable to mathematically rigorous study, it is likely that chemistry will also become an alert and demanding consumer of new mathematical results. The level of complexity of chemical problems is often very high, and modeling molecular behaviour and chemical reactions does require new mathematical approaches. Chemistry is witnessing an important shift in emphasis: simplistic models are no longer satisfactory, and more detailed mathematical understanding of complex chemical properties and phenomena are required. From theoretical chemistry and quantum chemistry to applied fields such as molecular modeling, drug design, molecular engineering, and the development of supramolecular structures, mathematical chemistry is an important discipline providing both explanations and predictions. JOMC has an important role in advancing chemistry to an era of detailed understanding of molecules and reactions.