An Improvised Cubic B-spline Collocation of Fourth Order and Crank–Nicolson Technique for Numerical Soliton of Klein–Gordon and Sine–Gordon Equations

IF 1.4 4区 综合性期刊 Q2 MULTIDISCIPLINARY SCIENCES
Saumya Ranjan Jena, Archana Senapati
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引用次数: 0

Abstract

This article examines the nonlinear hyperbolic Klein–Gordon equation (KGE) and sine–Gordon equation (SGE) with Crank–Nicolson and the finite element method (FEM) based on an improvised quartic order cubic B-spline collocation approach and explores their novel numerical solutions along with computational complexity. This work explains the parameters such as Euclidean error norms \({L}_{2}\), maximum absolute error \({L}_{\infty }\), and root-mean-square (RMS) error with computational time cost, experimental order of convergence (EOC), and three conservative laws \({I}_{1},{I}_{2},{I}_{3}\) of mass momentum and energy conservation, respectively. It is demonstrated that the method is unconditionally stable with the von-Neumann process and accurate to convergence of order O (\({h}^{4}+\Delta t)\). Finally, four test examples are investigated to support our assertion, and the experimental findings are compared to existing approaches using software tools like MATLAB and MATHEMATICA. 2D and 3D graphical representations of solutions are also presented and compared with the exact solution and results of others.

Klein-Gordon方程和sin - gordon方程数值孤子的四阶临时三次b样条配置和Crank-Nicolson技术
本文研究了基于Crank-Nicolson的非线性双曲Klein-Gordon方程(KGE)和正弦gordon方程(SGE),以及基于临时四次三次b样条配点法的有限元法(FEM),并探讨了它们的新颖数值解和计算复杂度。本文分别用计算时间成本、实验收敛顺序(EOC)和质量动量守恒和能量守恒三个守恒定律\({I}_{1},{I}_{2},{I}_{3}\)来解释欧几里得误差规范\({L}_{2}\)、最大绝对误差\({L}_{\infty }\)和均方根误差等参数。证明了该方法具有von-Neumann过程的无条件稳定性,并精确到O阶收敛(\({h}^{4}+\Delta t)\))。最后,通过四个测试实例来支持我们的断言,并使用MATLAB和MATHEMATICA等软件工具将实验结果与现有方法进行了比较。还给出了解的二维和三维图形表示,并与其他人的精确解和结果进行了比较。
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来源期刊
CiteScore
4.00
自引率
5.90%
发文量
122
审稿时长
>12 weeks
期刊介绍: The aim of this journal is to foster the growth of scientific research among Iranian scientists and to provide a medium which brings the fruits of their research to the attention of the world’s scientific community. The journal publishes original research findings – which may be theoretical, experimental or both - reviews, techniques, and comments spanning all subjects in the field of basic sciences, including Physics, Chemistry, Mathematics, Statistics, Biology and Earth Sciences
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