A novel hybrid variation iteration method and eigenvalues of fractional order singular eigenvalue problems

IF 1.7 3区化学 Q3 CHEMISTRY, MULTIDISCIPLINARY

Journal of Mathematical Chemistry Pub Date : 2024-07-15 DOI:10.1007/s10910-024-01651-3

Sarika Kumari, Lok Nath Kannaujiya, Narendra Kumar, Amit K. Verma, Ravi P. Agarwal

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引用次数: 0

Abstract

In response to the challenges posed by complex boundary conditions and singularities in molecular systems and quantum chemistry, accurately determining energy levels (eigenvalues) and corresponding wavefunctions (eigenfunctions) is crucial for understanding molecular behavior and interactions. Mathematically, eigenvalues and normalized eigenfunctions play crucial role in proving the existence and uniqueness of solutions for nonlinear boundary value problems (BVPs). In this paper, we present an iterative procedure for computing the eigenvalues ($\mu $) and normalized eigenfunctions of novel fractional singular eigenvalue problems,

$$\begin{aligned} D^{2\alpha } y(t) + \frac{k}{t^\alpha } D^\alpha y(t) + \mu y (t) =0,~~ 0< t<1,~~ 0< \alpha \le 1, \end{aligned}$$

with boundary condition,

$$y'(0)=0, ~~~~y(1)=0,$$

where $D^\alpha , D^{2\alpha }$ represents the Caputo fractional derivative, $k \ge 1$. We propose a novel method for computing Lagrange multipliers, which enhances the variational iteration method to yield convergent solutions. Numerical findings suggest that this strategy is simple yet powerful and effective.

Abstract Image

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新型混合变异迭代法和分数阶奇异特征值问题的特征值

为了应对分子系统和量子化学中复杂边界条件和奇异性带来的挑战，准确确定能级（特征值）和相应的波函数（特征函数）对于理解分子行为和相互作用至关重要。在数学上，特征值和归一化特征函数在证明非线性边界值问题（BVP）解的存在性和唯一性方面起着至关重要的作用。在本文中，我们提出了一种迭代过程，用于计算新型分数奇异特征值问题的特征值（$\mu $）和归一化特征函数，$$\begin{aligned}。D^{2\alpha } y(t) + \frac{k}{t^\alpha }D^{2\alpha } y(t) + \mu y (t) =0,~~ 0< t<1,~~ 0< \alpha \le 1, \end{aligned}$$边界条件为$y'(0)=0, ~~~~y(1)=0,$$ 其中 $D^\alpha , D^{2\alpha }$ 表示 Caputo 分数导数, $k \ge 1$.我们提出了一种计算拉格朗日乘数的新方法，该方法增强了变分迭代法，从而产生收敛解。数值结果表明，这一策略简单而强大有效。

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.