[Paper Review] Spectral Non-integer Derivative Representations and the Exact Spectral Derivative Discretization Finite Difference Method for the Fokker-Planck Equation

Dominic P. Clemence-Mkhope|arXiv (Cornell University)|Jun 4, 2021

Fractional Differential Equations Solutions3 references4 citations

TL;DR

This paper introduces spectral non-integer derivative representations and a novel exact spectral derivative discretization finite difference (ESDDFD) method for solving the Fokker-Planck equation. By leveraging self-sameness principles (SSP), the method accurately captures known physical behaviors—such as the Gibbs-Boltzmann distribution and Einstein-Stokes-Smoluchowski relation—exactly in discrete form, including for time-fractional cases, while unifying various non-integer derivatives as limit cases.

ABSTRACT

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.

Motivation & Objective

• To develop a finite difference method that exactly preserves known physical behaviors in the Fokker-Planck equation, including non-fractional and time-fractional cases.
• To unify various non-integer derivatives—such as conformable, Caputo, and Riemann-Liouville—within a single spectral derivative framework.
• To establish a self-sameness principle (SSP) as a foundational rule for rates of change in non-integer derivative representations.
• To demonstrate that the proposed ESDDFD method recovers exact physical laws without numerical approximation errors.

Proposed method

• The method is built on universal difference quotient representations derived from the self-sameness principle (SSP), which governs rates of change in non-integer derivative systems.
• A generalized derivative representation is introduced that reduces to known non-integer derivatives—local (conformable, fractal) and non-local (Caputo, Riemann-Liouville)—in appropriate limit cases.
• The ESDDFD method applies spectral discretization to the Fokker-Planck equation using these derivative representations, ensuring exact recovery of physical behaviors in the discrete model.
• The approach uses spectral methods to achieve high accuracy and stability, particularly for time-fractional Fokker-Planck equations.
• The method is validated by showing exact recovery of the Gibbs-Boltzmann distribution and the Einstein-Stokes-Smoluchowski relation in discrete form.
• The framework allows for the prediction of new physical behaviors not previously captured by standard finite difference schemes.

Experimental results

Research questions

• RQ1 Can a finite difference method be constructed that exactly preserves the Gibbs-Boltzmann distribution in the discrete solution of the Fokker-Planck equation?
• RQ2 How can non-integer derivatives of conformable, Caputo, and Riemann-Liouville types be unified under a single spectral derivative representation?
• RQ3 Does the ESDDFD method preserve the Einstein-Stokes-Smoluchowski relation in both non-fractional and time-fractional Fokker-Planck models?
• RQ4 What is the role of the self-sameness principle (SSP) in enabling exact discretization of non-integer derivative systems?
• RQ5 Can the proposed method predict new physical behaviors beyond known analytical solutions?

Key findings

• The ESDDFD method exactly recovers the Gibbs-Boltzmann distribution in the discrete solution of the Fokker-Planck equation, without numerical error.
• The method preserves the Einstein-Stokes-Smoluchowski relation in both non-fractional and time-fractional cases, confirming consistency with classical diffusion theory.
• The generalized derivative representation reduces to known non-integer derivatives—such as Caputo and Riemann-Liouville—as limiting cases, demonstrating unification across derivative types.
• The self-sameness principle (SSP) provides a universal rule for rates of change that underpins the exact discretization of non-integer derivatives.
• The method predicts new physical behaviors not previously captured by conventional finite difference schemes, extending the range of analytically exact numerical models.
• Spectral discretization ensures high accuracy and stability, particularly for time-fractional Fokker-Planck equations, by leveraging the intrinsic structure of the derivative representations.