Fractional Derivatives and Fractional ODEs

Why does rubber 'remember' its past shape? Why do some particles in a cell move slower than Brownian motion predicts? How can one describe a market where past prices influence future prices? The answer lies in derivatives of non-integer order, a tool Leibniz had already hinted at in 1695.

• **Biophysics**: diffusion of proteins in cell membranes is subdiffusive (alpha ~ 0.7), which controls the rate of biochemical reactions
• **Materials science**: viscoelastic materials (rubber, biological tissues, polymers) are described by fractional rheological equations with order alpha in (0, 1)
• **Finance**: fractional Brownian motion with H > 1/2 models long-range correlations in equity markets, where Hurst's R/S analysis measures 'memory'

Предварительные знания

• Nonlinear ODEs and Qualitative Analysis
• ODEs and PDEs in Machine Learning

Fractional derivatives: what is D^{1/2}?

The Lorenz attractor (1963) was discovered while modeling weather to 10⁻³ K sensitivity: forecasts lose accuracy after 10-14 days due to chaos. Ordinary derivatives have integer order: D^1 f = f', D^2 f = f'', D^3 f = f'''. What about order 1/2? Or pi? The idea of fractional calculus goes back to Leibniz (1695), who wrote to L'Hopital asking: 'What would d^{1/2}y/dx^{1/2} mean?' L'Hopital replied: 'It will be a paradox.' Leibniz answered: 'From this apparent paradox useful consequences will one day be drawn.'

The ordinary derivative f'(x) depends only on the values of f near x (local operation). The fractional derivative D^alpha f(x) depends on the values of f over the entire interval [a, x]: it is a **nonlocal** operation. Physical meaning: systems with 'memory' - their current state depends on the entire history, not just the present moment. Viscoelastic materials, anomalous diffusion in complex media, neural memory networks. Gamma function: Gamma(n) = (n-1)! for integer n; Gamma(1/2) = sqrt(pi), connecting to Gaussian integrals.

The Riemann-Liouville formula: rigorous definition

To define a fractional derivative rigorously for an arbitrary function (not just a monomial), one uses the **Riemann-Liouville formula**. The idea: a fractional derivative of order alpha is defined via a fractional integral of order (n - alpha), followed by an ordinary derivative of order n.

The kernel 1/(t-tau)^{1-alpha} in the fractional integral decays as a power law. This means: - The distant past (small tau) still influences the current value - The influence decays slowly (power law), not exponentially In ordinary ODEs 'memory' fades exponentially. In fractional ODEs it fades by a power law, characteristic of complex systems: polymers, porous media, neural memory networks.

Memory in fractional equations: nonlocality in time

Fractional ODEs describe **systems with memory**: the current state depends on the entire history, not only the present moment. This is a fundamental departure from ordinary ODEs and makes fractional calculus a powerful tool for modeling complex materials.

E_alpha(z) is a special function that generalizes e^z to fractional parameters: - E_1(z) = e^z - E_2(z) = cosh(sqrt(z)) - E_{1/2}(z) = e^{z^2} * erfc(-z) It interpolates between exponential and power-law decay. Applications in physics: dielectric relaxation (Cole-Cole law), viscoelastic materials, signal transmission in neural networks.

Anomalous diffusion, viscoelasticity, and fractals in time

Three major applications of fractional calculus: anomalous diffusion in complex media, viscoelastic materials (between elasticity and viscosity), and 'fractals in time' - processes with self-similar temporal structure.

In fractional equations the order alpha can be determined experimentally: **MSD**: measure the mean-squared displacement of a particle; the slope of log<x^2> vs. log(t) gives alpha **Dielectric spectroscopy**: the Cole-Cole law describes dielectric relaxation with fractional order **Rheology**: the frequency dependence of the shear modulus G(omega) ~ omega^alpha Fractional derivatives are therefore not merely a mathematical convenience but a direct description of real physical properties.

Key ideas

• **Fractional derivative D^alpha** generalizes ordinary derivatives to non-integer orders; main definitions: Riemann-Liouville and Caputo (more convenient for physical problems)
• **Nonlocality**: D^alpha f(t) depends on f over the entire interval [a, t]; the kernel (t-tau)^{alpha-n+1} encodes power-law memory
• **Mittag-Leffler function** E_alpha(-t^alpha) solves the fractional ODE; for large t it gives the power law t^{-alpha}, unlike the exponential of ordinary ODEs
• **Applications**: subdiffusion <x^2> ~ t^alpha (biophysics), viscoelasticity sigma = D^alpha epsilon (rheology), fractional Brownian motion (finance)

Related topics

Fractional derivatives connect analysis, physics, and modern applications:

• Stochastic differential equations — Fractional Brownian motion is an SDE with long-range memory (Hurst parameter H != 1/2)
• Heat equation — The fractional diffusion equation d^alpha u/dt^alpha = D * d^2u/dx^2 describes anomalous diffusion
• Laplace transform — Laplace techniques efficiently solve fractional ODEs: L{D^alpha f} = s^alpha F(s) minus initial terms

Вопросы для размышления

• The fractional derivative of a constant is not zero. This seems counterintuitive - how do one interpret it physically in the context of 'memory'?
• The Caputo derivative requires knowledge of f^(n)(tau) on [a, t]. What happens to the 'memory' if we change the lower limit of integration a?
• The Cole-Cole law in a dielectric: epsilon(omega) - epsilon_inf proportional to (1 + (i*omega*tau)^alpha)^{-1}. How does alpha relate to the 'molecular memory' of the material?

Связанные уроки

• calc-06-derivative-intro