What Are Fractional Differential Equations

From Integer Order to Fractional Order

Classical differential equations use integer-order derivatives. For example, the first derivative $\frac{du}{dt}$ describes velocity, and the second derivative $\frac{d^2u}{dt^2}$ describes acceleration. The orders of these derivatives are integers (1, 2, 3, ...).

Fractional Differential Equations (FDEs) generalize the order of derivatives to arbitrary real or complex numbers. For example, $\frac{d^{0.5}u}{dt^{0.5}}$ represents a half-order derivative, and $\frac{d^{1.7}u}{dt^{1.7}}$ represents a 1.7-order derivative. This enables equations to describe intermediate states between "pure diffusion" and "pure wave propagation," as well as systems with memory and nonlocal characteristics.

Basic Form

The general form of a fractional differential equation is:

Here $D^\alpha$ is the fractional-order derivative operator of order $\alpha$. Depending on the problem being studied, $D^\alpha$ can be a temporal fractional derivative (characterizing memory effects), a spatial fractional derivative (characterizing nonlocal interactions), or both.

Two Most Commonly Used Fractional Derivative Definitions

1. Riemann-Liouville (RL) Fractional Derivative

The RL derivative is defined by first performing a fractional-order integration and then taking an integer-order derivative:

where $n = \lceil \alpha \rceil$. The RL derivative has a concise Laplace transform form, but its initial conditions require the use of fractional integrals, making its physical interpretation less intuitive.

2. Caputo Fractional Derivative

The Caputo derivative reverses the order of operations, first taking an integer-order derivative and then performing a fractional-order integration:

The greatest advantage of the Caputo derivative is that initial conditions can use conventional integer-order derivatives (such as displacement and velocity), giving it a clear physical meaning. Therefore, in engineering and physics modeling, the Caputo derivative is far more commonly used than the RL derivative.

Why Fractional Order Is Needed

Memory Effect

The integral kernel $(t-s)^{-\alpha}$ of the temporal fractional derivative decays as a power law, meaning that the current state of the system depends on the entire history, not just the preceding moment. Integer-order derivatives care only about "the present," while fractional-order derivatives remember "everything in the past." This property makes fractional models particularly well-suited for describing: stress relaxation in viscoelastic materials, Warburg impedance in electrochemistry, and drug metabolism in biological systems.

Nonlocality and Anomalous Diffusion

Spatial fractional derivatives (such as the fractional Laplacian $(-\Delta)^s$) characterize spatial variations through global integration rather than local differences. This corresponds to particles making long-distance jumps (Lévy flights) rather than only short-range random walks (Brownian motion). As a result, fractional models can describe phenomena such as ultra-fast solute breakthrough in groundwater, jump diffusion in financial markets, and anomalous transport in plasmas.

An Intuitive Analogy

You can think of integer-order derivatives as "snapshots" -- recording only the instantaneous rate of change at the current moment. Fractional-order derivatives, by contrast, are like "time-lapse photography" -- superimposing the influence of all past moments onto the present with different weights. The closer the order $\alpha$ is to 0, the greater the weight of historical memory; the closer the order $\alpha$ is to 1, the closer it is to the classical local derivative.

Key Differences from Integer-Order Equations