Fractional-Order Differential Equations

Updated 27 December 2025

Fractional-Order Differential Equations (FODEs) are generalizations of classical differential operators using non-integer orders, effectively capturing memory and hereditary effects in dynamical systems.
Their formulation through Riemann–Liouville and Caputo definitions allows for classical initial conditions and explicit solutions using tools like Mittag–Leffler functions.
Advanced numerical and machine learning methods facilitate efficient simulation, parameter estimation, and analysis of complex FODEs in various scientific and engineering applications.

Fractional-Order Differential Equations (FODEs) generalize the classical notion of differentiation and integration to operators of arbitrary (non-integer) order, typically in the sense of Riemann–Liouville or Caputo. These nonlocal operators account for memory effects and anomalous dynamics in a variety of applications spanning physics, engineering, biology, and data-driven modeling. FODEs offer a flexible framework that often more accurately captures observed phenomena than their integer-order counterparts, especially in systems exhibiting hereditary, power-law, subdiffusive, or distributed-lag phenomena.

1. Mathematical Foundations and Definitions

The two predominant definitions of fractional derivatives for functions $y:[a,b]\to\mathbb{R}$ are the Riemann–Liouville (RL) and Caputo forms:

Riemann–Liouville Fractional Derivative of order $\alpha>0$ :

${}_a^{RL}D_t^{\alpha} y(t) = \frac{d^n}{dt^n}\left[ \frac{1}{\Gamma(n-\alpha)}\int_a^t (t-s)^{n-\alpha-1}y(s)ds \right], \quad n = \lceil\alpha\rceil$

Caputo Fractional Derivative of order $\alpha>0$ :

${}_a^CD_t^\alpha y(t) = \frac{1}{\Gamma(n-\alpha)}\int_a^t (t-s)^{n-\alpha-1}y^{(n)}(s)ds$

where $y^{(n)}$ denotes the $n$ -th classical derivative. Caputo derivatives are preferred in many physical applications as they admit standard integer-order initial conditions.

For fractional integrals: ${}_a I_t^{\alpha} y(t) = \frac{1}{\Gamma(\alpha)}\int_a^t (t-s)^{\alpha-1} y(s) ds$

These definitions nest classical calculus: as $\alpha \to n$ , the derivatives converge to the $n$ -th integer-order derivative. The nonlocal structure means that, for $0<\alpha<1$ , the rate of change at time $t$ depends on the entire past history of $y(s)$ , weighted by a singular power-law kernel $(t-s)^{-\alpha}$ , reflecting physical memory and hereditary effects (Almeida et al., 2015, Luchko, 2022).

2. Analytical Theory: Well-posedness, Solution Formulas, and Asymptotics

FODE initial value problems can be recast as Volterra integral equations, leveraging the equivalence: ${}_a^C D_t^{\alpha} y(t) = f(t, y(t)),\; y(a) = y_a \iff y(t) = y_a + \frac{1}{\Gamma(\alpha)}\int_a^t (t-s)^{\alpha-1}f(s, y(s))ds$ There is a well-established theory for existence and uniqueness of solutions under mild conditions (e.g., Lipschitz continuity of $f$ ) (Almeida et al., 2015, Luchko, 2022).

Many canonical FODEs admit explicit solutions in terms of Mittag–Leffler functions. For example, the scalar linear Caputo FODE

${}_0^C D_t^\alpha y(t) = \lambda y(t),\quad y(0)=y_0$

has the solution

$y(t) = y_0 E_\alpha(\lambda t^\alpha),\qquad E_\alpha(z) = \sum_{k=0}^\infty \frac{z^k}{\Gamma(\alpha k + 1)}$

Mittag–Leffler functions generalize the exponential and govern the sub-exponential, power-law late-time asymptotics characteristic of fractional kinetics. The precise decay exponent identifies the fractional order; e.g., for the solution $u(t) = E_\beta(r t^\beta)$ ,

$\beta = -\lim_{t \to \infty} \frac{t u'(t)}{u(t)}$

This provides a practical approach for inferring exponents from data (D'Ovidio et al., 2017, Górska et al., 2017).

The operational calculus and projector formalism for RL-type operators enable explicit solution formulas for multi-term linear FODEs using convolution series with generalized kernels, encompassing both initial conditions and inhomogeneous terms (Luchko, 2022).

3. Numerical Methods for FODEs

Numerical simulation and parameter identification in FODEs introduce significant new challenges due to nonlocality and weak singularities in convolution kernels.

3.1 Classical and Modern Discretization

Product-integration, Predictor–Corrector, and Adams-type Methods: Discretize the integral form of the FODE, using explicit or semi-implicit quadrature with accurately constructed weights, achieving $O(h^{\min\{1+\alpha,2\}})$ convergence for Caputo derivatives. FFT techniques can reduce computational cost from $O(N^2)$ to $O(N\log N)$ (Khalighi et al., 2022).
L1, L1-2 Schemes on Uniform/Nonuniform Meshes: L1 schemes provide error $O(h^{2-\alpha})$ and can be generalized to variable time steps for nonuniform grids, with strong discrete maximum principles and sharp Grönwall inequalities controlling numerical stability and algebraic decay (Feng et al., 4 Jan 2024).
Local Discontinuous Galerkin (LDG) Methods: Upwind LDG for FODEs provides element-by-element, stable high-order accuracy in $L^2$ and superconvergence at element interfaces, especially efficient for problems involving Mittag–Leffler function evaluation (Deng et al., 2014).
Fractional Runge–Kutta (FORK) Methods: Generalizations of explicit/implicit RK to FODEs using the Caputo Taylor formula enable multi-stage, higher $\alpha$ -order accuracy. Implicit FORK is A-stable, while explicit FORK faces stability barriers (Ghoreishi et al., 2022).
Second-Order $\theta$ -Methods: Fractional BDF, trapezoidal, and Newton–Gregory-type schemes deploying generating functions and correction terms achieve uniform global $O(h^2)$ accuracy and $A$ -stability across a range of fractional orders (Yin et al., 2019).
Integer-Order Expansion: Methods integrating by parts and expanding the Caputo kernel allow rewriting the fractional operator in terms of integer derivatives plus auxiliary variables, reducing FODEs to extended ODE systems suitable for standard ODE solvers (Almeida et al., 2015).

3.2 Variable-Order Methods

For variable-order FODEs, the Caputo derivative is discretized at each time step with $\alpha(t_n)$ , using nonuniform kernel weights and matrix update equations. Katzikadelis' method achieves $O(h^{1-\min_t\alpha(t)})$ accuracy for both linear and nonlinear variable-order FODEs, with the ability to handle variable coefficients and nonlocal memory kernels (Katsikadelis, 2018, Cui et al., 20 Mar 2025).

3.3 Discretization for Incommensurate Systems

Fractional systems may have vector-valued unknowns $\boldsymbol{x}(t)$ with componentwise, possibly different orders (commensurate or incommensurate). Modern solvers (e.g., FdeSolver) offer both predictor–corrector and implicit Newton solvers for such high-dimensional systems (Khalighi et al., 2022).

4. Data-Driven and Machine Learning Approaches

Recent advances integrate FODEs with neural networks, both for learning complex memory kernels from data and for using neural architectures to approximate solutions to FODEs.

Parameter Estimation from Data: Optimization frameworks leverage least-squares fitting of both fractional order $\alpha$ and system parameters $\theta$ to minimize data misfit, as in population, pharmacokinetic, and engineering models. Allowing $\alpha$ to vary provides substantial improvements in model fit where memory effects are pronounced (Almeida et al., 2015).
Sparse Regression Discovery: Alternating optimization frameworks combine neural network-based denoising, Gaussian–Jacobi quadrature for fractional derivative computation, and sparse regression to infer both structure and parameters (fractional orders, coefficients) directly from time-series and spatial data, demonstrating robust recovery even under substantial noise (Yu et al., 5 Dec 2024).
Neural FDEs: Neural architectures replace the integer-order time derivative in continuous-depth networks with a fractional derivative, where the order $\alpha$ may be learned as a parameter or function. Neural FDEs outperform neural ODEs for data with explicit memory dependence, albeit at higher computational cost due to $\mathcal{O}(N^2)$ memory requirements; adjoint methods can reduce this overhead (Coelho et al., 5 Mar 2024, Kang et al., 20 Mar 2025).
Variable-Order Neural Nets: Neural Variable-Order FDE (NvoFDE) models learn both the system state and feature-dependent fractional order dynamically, enhancing expressivity and performance in feature-adaptive settings (e.g., graph representation learning) (Cui et al., 20 Mar 2025).
Physics-Informed Neural Networks: Combined L1-discretization and Gaussian quadrature enable PINN-style architectures (deepFDEnet) to approximate FODE, FPDE, and integrodifferential equation solutions with high accuracy, with full backpropagation through mesh-free representations (Firoozsalari et al., 2023).

5. Application Domains and Physical Modeling

Fractional models accurately describe systems with:

Memory/Dissipative Phenomena: Population growth with demographic memory, anomalous subdiffusion, viscoelasticity, and blood alcohol dynamics are better fit by FODEs compared with classical ODEs, as evidenced by significant reductions in residual misfit (Almeida et al., 2015).
Electrical Circuits: Fractionalization of constitutive relations in Maxwell's equations leads to RL/Caputo FODE models for RCL circuits, preserving classical initial conditions and fitting frequency response anomalies (e.g., constant-phase elements, Cole impedance) (Moreles et al., 2016).
Subdiffusive and Allen–Cahn Equations: FODE-based discretizations for subdiffusion and phase-field models yield robust algebraic decay and uniform-in-time bounds, verified by discrete Grönwall inequalities on nonuniform meshes (Feng et al., 4 Jan 2024).
Multi-Term, Arbitrary-Order, and Nonstandard FODEs: Mikusiński-type operational calculus with kernel-convolution allows explicit representation and solution of single- and multi-term FODEs of arbitrary order and kernel, including general singular initial conditions (Luchko, 2022).

6. Stability, Asymptotics, and Periodicity

Decaying Modes: Stable solutions of FODEs typically decay algebraically as $t^{-\alpha}$ , rather than exponentially, with the decay exponent directly tied to the fractional order. Analytical and numerical schemes exploit the asymptotics of Mittag–Leffler functions for parameter inference and stability assessment (D'Ovidio et al., 2017, Górska et al., 2017).
Superconvergence: In LDG and certain finite element schemes, superconvergent rates are obtained at mesh endpoints due to the regularity of the underlying solution and the structure of the fractional kernel (Deng et al., 2014).
Periodic Solutions, Floquet Theory, and Hill Matrix: Classical Caputo-type FODEs do not admit truly periodic solutions due to infinite memory. Liouville–Weyl-type models, where fractional derivatives act from $t=-\infty$ , can have exact periodic orbits. However, extensions of Floquet theory using the Hill matrix characterize only exponentially growing instabilities, not algebraically decaying stable modes, reflecting fundamental limitations of exponential-based modal analysis in infinite-memory systems (Haacker et al., 29 Sep 2025).

7. Extensions: Generalized and Variable-Order FODEs

Generalized (Kernels/Operators): The development of kernels $(K,k)$ and associated operational calculus allows FODEs of arbitrary generalized order, closed forms for the projector, and characterizations of the null space, supporting natural non-integer initial data and broadening the class of treatable processes—including distributed and variable memory (Luchko, 2022).
Variable Order: Both deterministic and machine-learned variable-order FODEs support adaptive memory, critical for modeling systems where the memory effect itself evolves (e.g., inhomogeneous media, changing physical regimes) (Katsikadelis, 2018, Cui et al., 20 Mar 2025).