Skip to content
Oakfield Operator Calculus Function Reference Site

Fractional Integrals

📐 Definition

Section titled “📐 Definition”

For α>0\alpha > 0 and locally integrable ff on [0,T][0,T], the Riemann–Liouville fractional integral is

Itαf(t)=1Γ(α)∫0t(t−τ)α−1f(τ) dτI_t^{\alpha} f(t) = \frac{1}{\Gamma(\alpha)} \int_{0}^{t} (t-\tau)^{\alpha-1} f(\tau)\,d\tau

Domain and Codomain

Section titled “Domain and Codomain”

Defined for f∈Lloc1([0,T])f \in L^1_{\text{loc}}([0,T]); output is continuous on (0,T](0,T] and inherits weak singularity near t=0t=0 depending on α\alpha.

⚙️ Key Properties

Section titled “⚙️ Key Properties”

Linearity and Semigroup Property

Section titled “Linearity and Semigroup Property”
ItαItβf=Itα+βfI_t^{\alpha} I_t^{\beta} f = I_t^{\alpha+\beta} f

Laplace Transform

Section titled “Laplace Transform”

For suitable ff,

L{Itαf}(s)=s−α L{f}(s)\mathcal{L}\{I_t^{\alpha} f\}(s) = s^{-\alpha}\,\mathcal{L}\{f\}(s)

🎯 Special Cases and Limits

Section titled “🎯 Special Cases and Limits”
  • As α→0+\alpha\to 0^+, Itαf→fI_t^{\alpha} f \to f (in an appropriate sense).
  • For integer nn, ItnI_t^{n} reduces to nn-fold integration.
  • The kernel (t−τ)α−1(t-\tau)^{\alpha-1} decays algebraically in time difference, producing long memory effects.

🔗 Related Functions

Section titled “🔗 Related Functions”

Fractional derivatives invert fractional integrals (up to boundary conditions). Temporal memory kernels generalize the weighting beyond pure power laws.

Usage in Oakfield

Section titled “Usage in Oakfield”

Oakfield’s current fractional-time support is implemented via discrete approximations:

  • fractional_memory operator applies a finite-history fractional derivative/integral-style accumulator using Grünwald–Letnikov-like coefficients and a rolling history buffer.
  • subordination integrator implements a quadrature-based subordination scheme (time-change) that produces fractional-time behavior without an explicit convolution kernel.

Historical Foundations

Section titled “Historical Foundations”

📜 Fractional Calculus

Section titled “📜 Fractional Calculus”

Fractional integration and differentiation extend integer-order calculus to non-integer orders. Riemann–Liouville definitions provide one of the classical analytic formulations used in PDEs and memory models.

🌍 Modern Perspective

Section titled “🌍 Modern Perspective”

Fractional integrals serve as canonical power-law memory operators, connecting convolution kernels, Laplace multipliers, and anomalous transport models.

📚 References

Section titled “📚 References”
  • Podlubny, Fractional Differential Equations
  • Kilbas, Srivastava, Trujillo, Theory and Applications of Fractional Differential Equations