arXiv 0906.06762009-06math-phmath.MP

R^n中分形曲线的微积分

Calculus on Fractal Curves in R^n

Abhay Parvate, Seema Satin, A. D. Gangal

备注：: 23, 5 figures

摘要：

本文提出了一种新的分形曲线（如冯·科赫曲线）的计算方法.定义了沿分形曲线F的类黎曼积分，称为F-α-积分，其中α是F的维数。还定义了沿分形曲线的导数，称为F-α-导数。质量函数是曲线上的一个类似测量的算法量，在公式中起着核心作用。提出了一种计算质量函数的合适算法，以强调算法方面。这种微积分的几个方面保留了普通微积分的许多简单性。我们建立了一个共轭之间的这种微积分和普通微积分的实数线。证明了F-α-积分和F-α-导数分别与黎曼积分和普通导数共轭。事实上，它们可以用普通微积分和共轭中的相应算子来计算。在F上构造了Sobolev空间，推广了F-α-可微性.最后，我们通过一个沿分形路径吸收的例子来说明该框架在模型制作中的实用性。

英文摘要：

A new calculus on fractal curves, such as the von Koch curve, is formulated. We define a Riemann-like integral along a fractal curve F, called F-alpha-integral, where alpha is the dimension of F. A derivative along the fractal curve called F-alpha-derivative, is also defined. The mass function, a measure-like algorithmic quantity on the curves, plays a central role in the formulation. An appropriate algorithm to calculate the mass function is presented to emphasize algorithmic aspect. Several aspects of this calculus retain much of the simplicity of ordinary calculus. We establish a conjugacy between this calculus and ordinary calculus on the real line. The F-alpha-integral and F-alpha-derivative are shown to be conjugate to the Riemann integral and ordinary derivative respectively. In fact they can thus be evaluated using the corresponding operators in ordinary calculus and conjugacy. Sobolev Spaces are constructed on F, and F-alpha- differentiability is generalized. Finally we touch upon an example of absorption along fractal path to illustrate the utility of the framework in model making.

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英文摘要：

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