Science & Theory๏ƒ

HPFRACC is built on a rigorous mathematical foundation, enabling its use in high-stakes scientific research.

1. Scientific Applications๏ƒ

Computational Physics๏ƒ

  • Fractional Diffusion: Modeling anomalous transport in porous media or plasma physics.

  • Viscoelasticity: Simulating materials with both fluid and solid properties using fractional-order stress-strain relations.

  • Fractional Oscillators: Analyzing systems with non-local damping effects.

Biophysics๏ƒ

  • Protein Folding: Modeling the kinetics of conformational changes with memory.

  • Membrane Transport: Simulating sub-diffusive movement of ions through cellular membranes.

  • Pharmacokinetics: Modeling drug concentration decay with anomalous clearance rates.

2. Mathematical Foundations๏ƒ

Fractional calculus generalizes derivatives and integrals to any real order $alpha$.

Key Definitions๏ƒ

  • Gamma Function: \(\Gamma(z) = \int_0^{\infty} t^{z-1} e^{-t} dt\)

  • Riemann-Liouville Derivative:

    \[D^\alpha_{RL} f(t) = \frac{1}{\Gamma(n-\alpha)} \frac{d^n}{dt^n} \int_0^t (t-\tau)^{n-\alpha-1} f(\tau) d\tau\]

  • Caputo Derivative:

    \[D^\alpha_C f(t) = \frac{1}{\Gamma(n-\alpha)} \int_0^t (t-\tau)^{n-\alpha-1} f^{(n)}(\tau) d\tau\]

For a comprehensive derivation of all implemented operators, see the repository source docs/mathematical_theory.md (Markdown companion; not all auxiliary docs are in the default Sphinx toctree).

3. Numerical Stability & Accuracy๏ƒ

HPFRACC implements several strategies to ensure scientific validity:

  • Error Analysis: Built-in tools to compare numerical results against analytical solutions for common functions ($t^k$, $sin(t)$, etc.).

  • Convergence Monitoring: Automatic tracking of numerical error relative to step size $h$.

  • Intelligent Backend Choice: Backends are selected not just for speed, but for numerical precision (e.g., using JAX for high-precision spectral operations).