Riemann-Liouville Fractional Derivative

📐 Definition

Let $\alpha > 0$ and let $n = \lceil \alpha \rceil$ . The Riemann–Liouville fractional derivative of order $\alpha$ of a function $f$ is defined by

(D_{a+}^{\alpha} f)(x) = \frac{d^n}{dx^n} \left[ \frac{1}{\Gamma(n-\alpha)} \int_a^x (x-t)^{n-\alpha-1} f(t)\,dt \right]

where $\Gamma$ denotes the Gamma function. This definition generalizes integer-order differentiation by composing an ordinary derivative with a fractional integral.

Domain and Codomain

The Riemann–Liouville derivative is defined for functions $f$ that are locally integrable on $(a,\infty)$ and sufficiently regular for the integral and derivative to exist. The operator maps such functions to real- or complex-valued functions on $(a,\infty)$ .

⚙️ Key Properties

Linearity

The Riemann–Liouville derivative is linear:

D_{a+}^{\alpha}(af + bg) = a\,D_{a+}^{\alpha}f + b\,D_{a+}^{\alpha}g

for scalars $a,b$ .

Reduction to Integer-Order Derivatives

If $\alpha = n \in \mathbb{N}$ , then

D_{a+}^{n} f = \frac{d^n f}{dx^n}

recovering the classical derivative.

Power Functions

For $\beta > -1$ ,

D_{a+}^{\alpha}(x-a)^{\beta} = \frac{\Gamma(\beta+1)}{\Gamma(\beta-\alpha+1)}(x-a)^{\beta-\alpha}

This identity highlights the central role of the Gamma function in fractional calculus.

Nonlocality

The value of $D_{a+}^{\alpha} f(x)$ depends on the values of $f(t)$ for all $t \in [a,x]$ . This nonlocality encodes memory effects absent in integer-order differentiation.

Semigroup Property (Restricted)

For suitable functions and parameters,

D_{a+}^{\alpha} D_{a+}^{\beta} f = D_{a+}^{\alpha+\beta} f

holds under additional regularity assumptions. In general, the semigroup property does not hold without constraints.

🎯 Special Cases

Fractional Integral

For $0 < \alpha < 1$ , the corresponding Riemann–Liouville fractional integral is

(I_{a+}^{\alpha} f)(x) = \frac{1}{\Gamma(\alpha)} \int_a^x (x-t)^{\alpha-1} f(t)\,dt

The fractional derivative is obtained by differentiating this integral.

Constant Functions

For a constant function $f(x)=1$ ,

D_{a+}^{\alpha} 1 = \frac{(x-a)^{-\alpha}}{\Gamma(1-\alpha)}

Unlike the Caputo derivative, the Riemann–Liouville derivative of a constant is generally nonzero.

The Riemann–Liouville derivative is closely related to:

the Riemann–Liouville fractional integral,
the Caputo fractional derivative,
convolution operators with power-law kernels.

These operators form the core of classical fractional calculus.

Usage in Oakfield

Oakfield implements fractional-time behavior in two concrete places:

Fractional memory operator: fractional_memory maintains a rolling history buffer and applies a finite-memory fractional-derivative approximation (Grünwald–Letnikov–style coefficients) with scaling like $dt^{-\alpha}$ ; it operates component-wise even for complex fields (by treating storage as a flat double buffer).
Subordination integrator: the subordination integrator approximates fractional-time evolution by quadrature over a stretched-exponential kernel $\exp(-s^\alpha)$ and repeatedly evaluating the context drift in a side-effect-free “drift mode”.

These implementations are practical, discrete approximations; they capture “long memory” effects without requiring an exact closed-form Riemann–Liouville integral kernel.

Historical Foundations

📜 Origin (19th Century)

Bernhard Riemann introduced fractional integration while studying trigonometric series. Joseph Liouville later formalized fractional differentiation and extended the theory systematically.

🔬 Modern Development

Riemann–Liouville operators became foundational tools in fractional calculus, influencing probability theory, viscoelasticity, and anomalous transport models.

🌍 Modern Perspective

Today, the Riemann–Liouville derivative is viewed as a canonical fractional operator, particularly suited for analytic investigations and integral-equation formulations.

📚 References

Riemann, Versuch einer allgemeinen Auffassung der Integration und Differentiation
Liouville, Mémoire sur le calcul des différentielles à indices quelconques
NIST Digital Library of Mathematical Functions (DLMF), §1.14