Distributed-Order Fractional Models

• Distributed-order models are a class of fractional operators that integrate over a continuum of orders weighted by a measure to capture complex memory effects.
• They effectively represent anomalous diffusion, heterogeneous viscoelasticity, and multiscale nonlocal interactions by superposing derivatives at different scales.
• Applications span viscoelasticity, control theory, numerical analysis, and machine learning, supported by well-posed solution representations and explicit Green functions.

Distributed-order models constitute a foundational class in the modern mathematical theory of complex systems, generalizing classical and constant-order fractional differential models by superposing derivatives or integrals over a continuous (or discrete) spectrum of orders weighted by a nonnegative measure or function. The resulting framework is highly effective in representing systems and media exhibiting multiscale memory, anomalous transport without a unique scaling exponent, heterogeneous viscoelasticity, or variable nonlocality, as evidenced by applications spanning viscoelastic mechanics, anomalous diffusion, control theory, elasticity, numerical analysis, and machine learning.

1. Mathematical Formulation and Operators

A distributed-order (DO) derivative replaces the fixed differentiation order $\alpha$ in a fractional operator with an integral over a range $[\alpha_{\min}, \alpha_{\max}]$ weighted by a density or Radon measure. For a sufficiently smooth function $u(t)$,

$$D_t^{(\mu)}u(t) = \int_{\alpha_{\min}}^{\alpha_{\max}} D_t^{\alpha}u(t)\, \mu(d\alpha),$$

where $D_t^{\alpha}$ denotes either the Caputo or Riemann–Liouville fractional derivative of order $\alpha$ and $\mu$ is a nonnegative Borel measure, often taken as absolutely continuous with density $c(\alpha)$ so that $\mu(d\alpha)=c(\alpha)\,d\alpha$ (Broucke et al., 2022, Ferrás et al., 2022, 0912.2521). This operator superposes the memory effects associated with each $\alpha$, and by tuning $c(\alpha)$ (the "weighting function") captures a continuum of relaxation, creep, or anomalous diffusion behaviors.

Analogous constructions occur in space, e.g., using the Riesz–Riemann–Liouville fractional integral in nonlocal elasticity (Ding et al., 2021), and in material derivatives encoding both space and time in anomalous transport (Magdziarz et al., 2015).

2. Distributed-Order Evolution Equations

Distributed-order fractional equations generalize classical PDEs and FDEs by replacing a single temporal or spatial derivative with a DO operator: $$\int_{\alpha_{\min}}^{\alpha_{\max}} \varphi(\alpha)\, D_t^\alpha u(x,t)\, d\alpha + \mathcal{L}_x u(x,t) = f(x,t),$$ where $\varphi(\alpha)$ is the order-distribution density and $\mathcal{L}_x$ is a spatial (possibly nonlocal and distributed-order) operator (Samiee et al., 2018). Such models admit well-posedness and can interpolate between normal, fractional, and ultraslow dynamics by varying $\varphi$. The Laplace transform of a DO Caputo derivative yields

$$\mathcal{L}\{D_t^{(\mu)}u\}(s) = \Big(\int_{\alpha_{\min}}^{\alpha_{\max}} s^\alpha \mu(d\alpha) \Big) \tilde{u}(s).$$

This facilitates explicit solution representations and underpins spectral decomposition approaches (0912.2521).

In viscoelasticity, the distributed-order Maxwell model replaces classical or fractional springpot elements with DO elements, resulting in constitutive laws: $$\sigma(t) = \int_{0}^{1} c(\alpha)\, D_t^{\alpha} \gamma(t)\, d\alpha,$$ where the complex modulus is $G^*(s) = \int_0^1 c(\alpha)\, s^\alpha d\alpha$ (Ferrás et al., 2022).

3. Physical Interpretation and Stochastic Foundations

DO operators capture multiscale and heterogeneous memory phenomena not representable by single-exponent models. In distributed-order diffusion or transport equations, the underlying stochastic process is typically a random walk or Lévy walk subordinated by an ultra-slow subordinator mixing all stable indices:

• In (Magdziarz et al., 2015), the DO material derivative arises as the scaling limit of a coupled process with random waiting-time exponents whose law is $p(\beta)$. The resulting limit process is non-Markovian, incorporating mixed power-law waiting times to model ultraslow or retarding subdiffusion.
• The distribution over $\alpha$ describes the ensemble of effective 'memory kernels' or 'relaxation rates' present in heterogeneous materials, random environments, or architected media (Ding et al., 2021).

A prominent application is in anomalous diffusion without a single scaling exponent, retarding subdiffusion, or ultraslow logarithmic diffusion, where the mean-squared displacement (MSD) can exhibit time-dependent exponents or logarithmic growth depending on $\mu(\alpha)$ (Eab et al., 2010).

4. Applications in Continuum Mechanics, Control, and Machine Learning

Distributed-order models are widely utilized in:

• Viscoelasticity and Rheology: The generalized distributed-order Maxwell model realizes, via appropriate choices of $c(\alpha)$, a vast spectrum of relaxation moduli and creep compliances. Canonical choices such as Dirac, uniform, or exponential $c(\alpha)$ yield, respectively, power-law, logarithmic, or mixed intermediate decay in moduli and compliance, providing close empirical fits to complex-fluid data (Ferrás et al., 2022). The framework unifies integer-order (Maxwell), fractional, and distributed-order rheologies.
• Nonlocal Elasticity and Multiscale Mechanics: Distributed-order elasticity generalizes constant-order models to systems with spatially varying or scale-dependent nonlocality. The operator's kernel $\kappa(\alpha)$ quantifies the contribution of each spatial scale, enabling accurate modeling of architected materials, layered composites, or variable-impurity junctions (Ding et al., 2021). Mappings between the continuum DO theory and discrete mass–spring lattices enable physical interpretability and numerical validation.
• Optimal Control: Control systems involving DO derivatives admit a Pontryagin Maximum Principle in which the adjoint (co-state) equations and transversality conditions become distributed-order, reflecting the extended memory effects in the system's dynamics (Ndairou et al., 2020). This setting generalizes both classical and fractional control, and sufficiency conditions under convexity extend directly.
• Graph Neural Networks (GNNs): The DRAGON framework utilizes DO fractional derivatives in continuous GNNs, with a learnable distribution over orders, significantly outperforming integer- and single-order baselines for long-range, homophilic, heterophilic, and classification tasks. The flexible superposition of orders adapts to arbitrary non-Markovian memory kernels in graph convolutions (Zhao et al., 8 Nov 2024).

5. Well-posedness, Stability, and Numerical Analysis

• Well-posedness: Under regularity and integrability conditions on the weighting measure $\mu$ (such as $\int_0^1 \mu(d\beta)/(1-\beta) < \infty$), distributed-order evolution equations possess unique classical or strong solutions, both in bounded and unbounded domains (0912.2521, Broucke et al., 2022).
• Thermodynamics and Constitutive Restrictions: For viscoelastic media, thermodynamical admissibility of the constitutive law requires positivity of storage and loss moduli, which translates into nonnegativity criteria on the signed measures associated with the order distributions $\mu_\sigma, \mu_\varepsilon$ (Broucke et al., 2022).
• Numerical Schemes: Fast, stable Petrov–Galerkin spectral methods have been constructed for distributed-order PDEs, requiring the definition of distributed Sobolev spaces and equivalent norms adapted to the order-distribution (Samiee et al., 2018). Weak formulations, suitable for variational and spectral techniques, involve bilinear forms with DO derivatives in trial and test spaces.

6. Solution Representations and Asymptotic Analysis

Distributed-order evolution equations often admit representations via Laplace or Fourier analysis:

• Eigenfunction expansions: For bounded domains, solutions decompose into series over the eigenfunctions of the spatial operator, with time-dependent coefficients satisfying scalar DO ODEs (0912.2521).
• Stochastic representations: Solutions are expressible as expectations over Markov processes subordinated by DO subordinators—e.g., Feynman–Kac-type formulas involving non-Markovian inverse subordinator times (0912.2521, Magdziarz et al., 2015).
• Explicit Green functions: In special cases (e.g., double-delta, uniform, or power-law $\mu$), time-domain Green functions and mean-square asymptotics can be computed, typically involving Mittag–Leffler, Meijer G, or exponential-integral functions (Eab et al., 2010, Ferrás et al., 2022).
• Ultraslow and Retarding Diffusion: For continuous measures $\mu$, long-time behavior can exhibit logarithmic or stretched power-law decay, with the effective scaling exponent depending on moment integrals of $\mu$ (Eab et al., 2010).

7. Parameter Effects, Model Selection, and Physical Interpretability

The choice of the distribution $\mu$, or the corresponding density $c(\alpha)$ or $\kappa(\alpha)$ (for spatial DO), fundamentally determines the transient and asymptotic behaviors of the system:

• Dirac measures yield classical or single-exponent fractional behavior.
• Uniform or linear weights realize ultraslow diffusion, logarithmic decay, or anomalous retardation.
• Power-law or exponentially biased weights enable the modeling of complex spectra of relaxation or memory times (Ferrás et al., 2022, Eab et al., 2010).

In applications, physical interpretability is achieved by relating $\mu$ to distributions of memory kernels, spatial interactions, or waiting-time exponents in random processes. Distributed-order models thus deliver a unifying mathematical language for media and phenomena characterized by heterogeneous, multiscale, and history-dependent dynamics.