Numerical solution of one- and two-dimensional Hyperbolic Telegraph equation via Cubic–Quartic Hyperbolic B-Spline DQM: a statistical validity

IF 1.7 3区化学 Q3 CHEMISTRY, MULTIDISCIPLINARY

Journal of Mathematical Chemistry Pub Date : 2024-07-24 DOI:10.1007/s10910-024-01652-2

Mamta Kapoor

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Abstract

In present research work, numerical approx. of one- and two-dimensional Hyperbolic Telegraph equations is fetched with aid of Modified Cubic and Quartic Hyperbolic B-spline Differential Quadrature Methods. Modified cubic B-spline is used in Differential Quadrature Method to find weighting coefficients for Method I. Modified Quartic Hyperbolic B-spline is utilized to attain weighting coefficients for Method II. After spatial discretization partial differential equations got reduced in the system of ODEs, which later on tackled with SSPRK43 regime. Total ten Examples are discussed to check the efficacy and robustness of the implemented method. For comparison of results, error norms are evaluated. Graphical presentation of the results is also provided. It got noticed that, in most of the cases, exact solutions and present numerical solutions were compatible. The present scheme is easy to implement and it is a better approach to solve some complex natured partial differential equations. The cubic Hyperbolic B-spline has produced much better errors than the Quartic Hyperbolic B-spline. The statistical validation of the parameters is also provided via generating the correlation matrix heatmap.

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通过三次四次双曲 B-样条 DQM 计算一维和二维双曲 Telegraph 方程的数值解法：统计有效性

在本研究工作中，借助修正三次双曲 B-样条微分正交法对一维和二维双曲 Telegraph 方程进行了数值近似计算。在微分正交法中使用了修正三次双曲 B-样条线，以找到方法 I 的加权系数。修正四次双曲 B-样条曲线用于方法 II 的加权系数。在空间离散化之后，偏微分方程被简化为 ODEs 系统，随后用 SSPRK43 机制加以解决。共讨论了十个实例，以检查所实施方法的有效性和稳健性。为了比较结果，对误差规范进行了评估。此外，还提供了结果的图形展示。我们注意到，在大多数情况下，精确解与目前的数值解是一致的。本方案易于实施，是解决某些性质复杂的偏微分方程的更好方法。三次双曲 B-样条线产生的误差比四次双曲 B-样条线好得多。此外，还通过生成相关矩阵热图对参数进行了统计验证。

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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.