Fractional Finite-Difference Kernels

📐 Definition

Fractional finite-difference kernels use generalized binomial coefficients to approximate derivatives of fractional order through discrete convolution (Grünwald–Letnikov type formulas).

One common family of weights is

w_k = (-1)^k \binom{\alpha}{k}, \qquad (\Delta_h^\alpha f)_n = \frac{1}{h^\alpha}\sum_{k=0}^\infty w_k\, f_{n-k}

Domain and Codomain

Defined on discrete grids (typically uniform index sets) with spacing $h>0$ . The output is a weighted sum representing a fractional-order derivative proxy, with long-range coupling through slowly decaying weights.

⚙️ Key Properties

Generalized Binomial Coefficients

For real/complex $\alpha$ and integer $k\ge 0$ ,

\binom{\alpha}{k} = \frac{\Gamma(\alpha+1)}{\Gamma(k+1)\Gamma(\alpha-k+1)}

Long-Range Influence

The coefficients $w_k$ decay polynomially with $k$ (for non-integer $\alpha$ ), capturing nonlocal influence over many grid points.

Integer Orders Recover Classical Stencils

For integer $\alpha=n$ , the series truncates and reduces to classical finite differences.

🎯 Special Cases and Limits

$\alpha=1$ gives the first-order backward difference: $\Delta f_n = f_n - f_{n-1}$ .
$\alpha=2$ yields the standard second-order stencil.
As $\alpha\to 0$ , $w_0\to 1$ and the remaining weights vanish, recovering the identity.

Binomial coefficients supply the weights. Finite differences are the integer-order special cases. Fractional derivatives and integrals are the continuous analogues of the same power-law kernel structure.

Usage in Oakfield

Oakfield implements fractional finite-difference-style kernels via its temporal memory operator:

fractional_memory operator applies a history-weighted stencil over past timesteps using generalized binomial coefficients, acting as a discrete approximation to fractional derivatives/integrals.
The kernel is finite-memory (bounded window) rather than truly infinite-range, which makes it practical for real-time simulation.

Historical Foundations

📜 Generalized Binomial Series

Generalized binomial coefficients extend the discrete combinatorial pattern to non-integer orders via $\Gamma$ , enabling fractional-order difference formulas that mirror fractional calculus in the continuum.

🌍 Modern Perspective

These kernels provide a simple discrete route to fractional operators, at the cost of nonlocality and long tails in the stencil.

📚 References

Podlubny, Fractional Differential Equations
LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations