Spectral Noninteger Derivative Representations and the Exact Spectral Derivative Discretization Finite Difference Method for the FokkerPlanck Equation
Abstract
Universal difference quotient representations are introduced for the exact selfsameness principles (SSP) as rules for rates of change introduced in [ClemenceMkhope, 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 noninteger derivatives as limit cases of such natural derivative measures; this is shown for some local derivatives of conformable, fractional, or fractal type and nonlocal derivatives of Caputo and RiemannLiouville type. The SSPinspired exact spectral derivative discretization finite difference method is presented for the FokkerPlanck nonfractional and timefractional equations; the resulting discrete models recover exactly some known behaviors predicted for the processes modeled, such as the GibbsBoltzmann distribution and the EinsteinStokesSmoluchowski relation; new ones are predicted.
 Publication:

arXiv eprints
 Pub Date:
 June 2021
 DOI:
 10.48550/arXiv.2106.02586
 arXiv:
 arXiv:2106.02586
 Bibcode:
 2021arXiv210602586C
 Keywords:

 Mathematics  Numerical Analysis;
 Mathematical Physics
 EPrint:
 21 pages