{"title":"Numerical treatment of singularly perturbed turning point problems with delay in time","authors":"Satpal Singh, Devendra Kumar, J. Vigo-Aguiar","doi":"10.1007/s10910-025-01707-y","DOIUrl":null,"url":null,"abstract":"<div><p>This paper proposes a uniformly convergent numerical method for a class of singularly perturbed turning point problems with a time-lag defined on a rectangular domain. We consider an interior repulsive turning point with odd multiplicity <span>\\(\\geqslant 1\\)</span>. Twin boundary layers arise in the proximity of endpoints of the spatial domain due to the presence of the perturbation parameter. Preliminary results such as minimum principle, stability estimate, and solution derivative bounds for the continuous problem applicable in the convergence analysis are presented. First, we employ the Crank–Nicolson scheme to semi-discretize the continuous problem in the time direction, and then the cubic <span>\\(\\mathscr {B}\\)</span>-spline functions on an appropriate Shishkin mesh are used to get a full discretization. The convergence analysis uses the maximum norm to obtain parameter-uniform error estimates. Three test problems are solved numerically to validate the theoretical results and confirm the scheme’s effectiveness.\n</p></div>","PeriodicalId":648,"journal":{"name":"Journal of Mathematical Chemistry","volume":"63 4","pages":"1035 - 1064"},"PeriodicalIF":2.0000,"publicationDate":"2025-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Mathematical Chemistry","FirstCategoryId":"92","ListUrlMain":"https://link.springer.com/article/10.1007/s10910-025-01707-y","RegionNum":3,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}
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Abstract
This paper proposes a uniformly convergent numerical method for a class of singularly perturbed turning point problems with a time-lag defined on a rectangular domain. We consider an interior repulsive turning point with odd multiplicity \(\geqslant 1\). Twin boundary layers arise in the proximity of endpoints of the spatial domain due to the presence of the perturbation parameter. Preliminary results such as minimum principle, stability estimate, and solution derivative bounds for the continuous problem applicable in the convergence analysis are presented. First, we employ the Crank–Nicolson scheme to semi-discretize the continuous problem in the time direction, and then the cubic \(\mathscr {B}\)-spline functions on an appropriate Shishkin mesh are used to get a full discretization. The convergence analysis uses the maximum norm to obtain parameter-uniform error estimates. Three test problems are solved numerically to validate the theoretical results and confirm the scheme’s effectiveness.
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.
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