THE INTEGRAL FORMULA FOR CALCULATINGTHE HAUSDORFF MEASURE OF SOME FRACTAL SETS

It is important to calculate the Hausdorff dimension and the Hausdorff mesure respect to this dimension for some fractal sets. By using the usual method of “Mass Distribution”, we can only calculate the Hausdorff dimension. In this paper, we will construct an integral formula by using lower inverse s-density and then use it to calculate the Hausdorff measures for some fractional dimensional sets.

Fractal Calculus and Analysis for Characterizing Geoanomalies Caused by Singular Geological Processes

Integral and differentiation are two mathematical operations in modern calculus and analysis which have been commonly applied in many fields of science.Integration and differentiation are associated and linked as inverse operation by the fundamental theorem of calculus.Both integral and differentiation are defined based on the concept of additive Lebesgue measure although various generations have been developed with different forms and notations.Fractals can be considered as geometry with fractal dimension(e.g.,non-integer)which no longer possesses Lebesgue additive property.Accordingly,the ordinary integral and differentiation operations are no longer applicable to the fractal geometry with singularity.This paper introduces a recently developed concept of fractal differentiation and integral operations.These operations are expressed using the similar notations of the ordinary operations except the measures are defined in fractal space or measures with fractal dimension.The calculus operations can be used to describe the new concept of fractal density,the density with fractal dimension or density of matter with fractal dimension.The concept and methods are also applied to interpret the Bouguer anomaly over the mid-ocean ridges.The results show that the Bouguer gravity anomaly depicts singularity over the mid-ocean ridges.The development of new calculus operations can significantly improve the accuracy of geodynamic models.

THE FLOW ANALYSIS OF FLUIDS IN FRACTAL RESERVOIR WITH THE FRACTIONAL DERIVATIVE

In this paper, fractional order derivative, fractal dimension and spectral dimension are introduced into the seepage flow mechanics to establish the flow models of fluids in fractal reservoirs with the fractional derivative. The flow characteristics of fluids through a fractal reservoir with the fractional order derivative are studied by using the finite integral transform, the discrete Laplace transform of sequential fractional derivatives and the generalized Mittag-Leffler function. Exact solutions are obtained for arbitrary fractional order derivative. The long-time and short-time asymptotic solutions for an infinite formation are also obtained. The pressure transient behavior of fluids flow through an infinite fractal reservoir is studied by using the Stehfest＇s inversion method of the numerical Laplace transform. It shows that the order of the fractional derivative affect the whole pressure behavior, particularly, the effect of pressure behavior of the early-time stage is larger The new type flow model of fluid in fractal reservoir with fractional derivative is provided a new mathematical model for studying the seepage mechanics of fluid in fractal porous media.