Counterexample of Local Fractional Order Chain Rule and Modified Definition of Local Fractional Order

• 作者: FAN Kai ZHOU Cunlong
• 来源期刊: 东华大学学报（英文版）
• 学科: 数学
• 年，卷（期）: 2020,（6）
• 页码范围: 521-525
• 页数: 5页
• 分类号: O172.1
• 语种: 英文
• DOI: 10.19884/j.1672-5220.202009060

摘要

Fractional calculus is a powerful tool for modeling nonlinear systems. It is necessary to discuss the basic properties of fractional order before solving a fractional order model. Using the formula of power function defined by local fractional derivative and the chain rule to calculate a compound function, the results are inconsistent. This shows that the chain rule of local fractional derivatives similar to classical calculus is suspicious, and fractional complex transformation based on the chain rule is also suspicious and needs further discussion. In order to overcome this inconsistency, an improved definition of local fractional derivative, which can be regarded as a fractal derivative, is proposed based on the results derived from the relationship between the mass function and the Hausdorff measure.