分形Gardner方程的广义变分原理

IF 3.3 3区数学 Q1 MATHEMATICS, INTERDISCIPLINARY APPLICATIONS

Fractals-Complex Geometry Patterns and Scaling in Nature and Society Pub Date : 2023-10-17 DOI:10.1142/s0218348x23501207

Kang-Jia Wang

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引用次数: 0

摘要

近年来，分形演算得到了更广泛的关注。分形变分原理在分形偏微分方程的分形行波理论中起着重要作用。本文首次利用半逆方法发展了分形Gardner方程的分形广义变分原理(GVP)。另一方面，我们也利用分形二尺度变换(FTST)从另一个维度域讨论并验证了分形GVP。提取的分形GVP通过分形空间中的能量形式显示出守恒规律，可用于分形孤波性质的探索。

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On the generalized variational principle of the fractal Gardner equation

The fractal calculus has gained more widespread attention in the last years. The fractal variational principle plays a major role in the fractal travelling wave theory of the fractal PDEs. This paper develops the fractal generalized variational principle (GVP) of the fractal Gardner equation by virtue of the Semi-inverse method (SIM) for the first time. On the other hand, we also discuss and verify the fractal GVP via the fractal two-scale transform (FTST) from another dimension field. The extracted fractal GVP shows the conservation laws through the energy form in the fractal space, and can be manipulated to explore the fractal solitary wave properties.

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来源期刊

Fractals-Complex Geometry Patterns and Scaling in Nature and Society 数学-数学跨学科应用

CiteScore

7.40

自引率

23.40%

发文量

319

审稿时长

>12 weeks

期刊介绍： The investigation of phenomena involving complex geometry, patterns and scaling has gone through a spectacular development and applications in the past decades. For this relatively short time, geometrical and/or temporal scaling have been shown to represent the common aspects of many processes occurring in an unusually diverse range of fields including physics, mathematics, biology, chemistry, economics, engineering and technology, and human behavior. As a rule, the complex nature of a phenomenon is manifested in the underlying intricate geometry which in most of the cases can be described in terms of objects with non-integer (fractal) dimension. In other cases, the distribution of events in time or various other quantities show specific scaling behavior, thus providing a better understanding of the relevant factors determining the given processes. Using fractal geometry and scaling as a language in the related theoretical, numerical and experimental investigations, it has been possible to get a deeper insight into previously intractable problems. Among many others, a better understanding of growth phenomena, turbulence, iterative functions, colloidal aggregation, biological pattern formation, stock markets and inhomogeneous materials has emerged through the application of such concepts as scale invariance, self-affinity and multifractality. The main challenge of the journal devoted exclusively to the above kinds of phenomena lies in its interdisciplinary nature; it is our commitment to bring together the most recent developments in these fields so that a fruitful interaction of various approaches and scientific views on complex spatial and temporal behaviors in both nature and society could take place.