Investigation of real-world second-order singular differential equations by optimal homotopy analysis technique

IF 1.7 3区 化学 Q3 CHEMISTRY, MULTIDISCIPLINARY
Randhir Singh, Prabal Datta, Vandana Guleria, Nirupam Sahoo
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引用次数: 0

Abstract

Extensive studies have investigated second-order singular differential equations to model various phenomena in astrophysics, reaction-diffusion processes, and electrohydrodynamics. However, finding numerical and analytical solutions for these problems with appropriate boundary conditions is challenging due to their inherent nonlinearity. Our current study explores singular second-order differential equations (SSODEs) with boundary conditions, specifically those modelling the distribution of heat sources in the human head and the steady-state temperature distribution in a vessel before a thermal explosion. The fundamental idea behind our approach is initially transforming the differential equation into an equivalent integral form, thereby circumventing the singular behaviour. Subsequently, the optimal homotopy analysis method is employed to scrutinize two distinct models, i.e., the heat conduction model, the thermal explosion model and the spherical catalyst equation. Further, a detailed convergence analysis is conducted in a Banach space framework to ensure the method’s reliability. The accuracy of the new approach is checked by considering various numerical examples with different values of thermogenesis heat production, the Biot number, and metabolic thermogenesis slope. It has been shown that the proposed approach qualitatively and quantitatively approximates the solutions with higher precision than the existing Adomian decomposition method.

用最优同伦分析技术研究现实世界二阶奇异微分方程
广泛的研究调查了二阶奇异微分方程来模拟天体物理学、反应扩散过程和电流体动力学中的各种现象。然而,由于这些问题固有的非线性,寻找具有适当边界条件的数值和解析解是具有挑战性的。我们目前的研究探索了具有边界条件的奇异二阶微分方程(SSODEs),特别是那些模拟人类头部热源分布和热爆炸前容器内稳态温度分布的方程。我们的方法背后的基本思想是首先将微分方程转化为等效积分形式,从而绕过奇异行为。随后,采用最优同伦分析方法对热传导模型、热爆炸模型和球形催化剂方程这两种不同的模型进行了研究。并在Banach空间框架下进行了详细的收敛性分析,保证了方法的可靠性。通过考虑不同产热值、Biot数和代谢产热斜率的数值实例,验证了新方法的准确性。结果表明,与现有的Adomian分解方法相比,该方法定性和定量地逼近解的精度更高。
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来源期刊
Journal of Mathematical Chemistry
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.
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