Fractal analysis of Hausdorff calculus and fractional calculus models

• 1、State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering，Hohai University, Nanjing 210098, China
• 2、College of Mechanics and Materials, Hohai University, Nanjing 210098, China

Abstract：This study clearly interprets the order of the Hausdorff derivative and fractional calculus from the fractal dimensionality and distinguishes the relationship and difference between these two calculus modeling approaches. This is the first time that the fractal geometry foundation of fractional calculus is quantitively clarified. We also provides the geometric interpretation of the Hausdorff derivative model's description of history-dependency process, namely, its dependency on initial time, in a direct comparison with the fractional derivative model. Based on the author's early work, this paper provides the detailed definition of the fractal distance, a non-Euclidean metric. The fundamental solution of the Huasdorff derivative diffusion equation is based on this fractal distance underlying popular statistical models of the stretched Gaussian distribution and the stretched exponential decay.

Keywords： Hausdorff derivative Hausdorff calculus fractional calculus non-Euclidean metric fractal distance Hausdorff distance fundamental solution