ar X iv : 0 90 6 . 06 76 v 1 [ m at h - ph ] 3 J un 2 00 9 Calculus on Fractal Curves in
نویسنده
چکیده
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 α-integral, where α is the dimension of F . A derivative along the fractal curve called Fα-derivative, is also defined. The mass function, a measurelike 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 Fα-integral and Fαderivative are shown to be conjugate to the Riemann integral and ordinary derivative respectively. In fact, they can thus be evalutated using the corresponding operators in ordinary calculus and conjugacy. Sobolev Spaces are constructed on F , and Fαdifferentiability is generalized using Sobolev like constructions. Finally we touch upon an example of a diffusion equation on fractal curves, to illustrate the utility of the framework.
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