Universal difference quotient representations are introduced for the exact self-sameness principles (SSP) as rules for rates of change introduced in [Clemence-Mkhope, D.P (2021, Preprint). The Exact Spectral Derivative Discretization Finite Difference (ESDDFD) Method for Wave Models. arXiv]. Properties are presented for the fundamental rule, a generalized derivative representation which is shown to yield some known non-integer derivatives as limit cases of such natural derivative measures; this is shown for some local derivatives of conformable, fractional, or fractal type and non-local derivatives of Caputo and Riemann-Liouville type. The SSP-inspired exact spectral derivative discretization finite difference method is presented for the Fokker-Planck non-fractional and time-fractional equations; the resulting discrete models recover exactly some known behaviors predicted for the processes modeled, such as the Gibbs-Boltzmann distribution and the Einstein-Stokes-Smoluchowski relation; new ones are predicted.
翻译:对精确自我比照原则(SSP)采用了通用的离差表示法,作为[Clemence-Mkhope,D.P(2021,Preprint))中引入的变革率规则。波形模型的精谱光谱衍生分异性细微差异法(ESDDFD)。为基本规则提出了属性,即普遍衍生衍生物代表法,表明它会产生某些已知的非内源衍生物,作为这类自然衍生物的极限;为卡普托和里曼-利乌维尔类型的符合性、分数型或分形型或分形型或非本地衍生物和非本地衍生物的本地衍生物展示了这种特征。 SSP为Fokker-Planck非反射和时间反射方程式提供的精确光谱分解有限差异法;由此产生的离子模型恢复了某些已知的模型,如Gibbbs-Boltzmann分布和爱因斯坦-斯托克-Smoluchowski关系;新模型预测了。