Fractional Multi-Order Problems in Complex Systems

• Fractional multi-order problems are mathematical models using non-integer derivatives of varying orders to capture memory effects and multiscale dynamics.
• These systems employ specialized techniques including vector-indexed Mittag-Leffler functions and operator semigroup theory to ensure well-posed analytic representations.
• Robust numerical discretizations and optimal control frameworks, such as Runge–Kutta schemes and Pontryagin principles, enable precise simulation and parameter identification in complex applications.

Fractional Multi-Order Problems

Fractional multi-order problems are dynamical systems, differential equations, or control frameworks in which fractional (non-integer) derivatives of differing orders—often incommensurate, state- or field-dependent, or coupled via auxiliary systems—appear in the governing evolution laws. These problems generalize constant-order and variable-order fractional models and are motivated by the rich memory and interaction effects arising in complex multi-scale and multi-physics systems, such as composite media, viscoelasticity, anomalous diffusion, and engineered systems with interacting heterogeneous components. The mathematical formulation, well-posedness, analytic representation, numerical methods, and control principles for such systems require extensions of classical fractional calculus, operator semigroup theory, and specialized discretization or optimization frameworks.

1. Mathematical Frameworks for Multi-Order Fractional Systems

Three main classes of multi-order problems are encountered:

1. Classical multi-term/multi-component systems: The system features several fractional derivatives of distinct orders in either single or coupled equations. For a vector state $U=(u_1,...,u_m)^T$, the archetype is

$$D_t^{\beta_j}u_j(t) = \sum_{k=1}^m f_{jk}(A)u_k(t) + H_j(t), \qquad 0<\beta_j\le 1,$$

with $f_{jk}(A)$ operators on a Banach space, allowing for both commensurate (all $\beta_j$ equal) and incommensurate cases (Umarov, 4 Feb 2024).

2. Dynamic-order (state-/auxiliary-driven order) systems: The local derivative order in the evolution equation depends on the time, space, and/or the output of other dynamic systems, representing memory modulation directly driven by coupled subsystems. A generic form is

${}^C D_{t}^{\alpha(y(t))}x(t) + B x(t) = f(t),$

with $\alpha(y)$ determined by another ODE or auxiliary process (Sun et al., 2011).

3. Bi-order and generalized multi-order kernels: Here, each term or kernel in the fractional operator involves multiple orders, e.g., using power-law and generalized Mittag-Leffler kernels acted upon simultaneously. Two-order derivatives such as those introduced by Atangana yield

$${}_{a}^{\!AC}D^{\alpha,\beta}f(x) = \frac{A(\beta)}{1-\beta} \int_a^x f^{(n)}(t) (x-t)^{n-\alpha-1} E_\beta\left(-\frac{\beta}{1-\beta} (x-t)^{\alpha+\beta}\right) dt,$$

allowing modeling of layered media or multiple mechanisms of anomalous relaxation (Atangana, 2016).

In all frameworks, the choice between Riemann–Liouville, Caputo, or other fractional derivatives (and the possibility of variable or vector-valued order) is central and directly affects analytical and numerical properties.

2. Analytic Representation, Existence, and Uniqueness

Analytic Solution Representations:

• Explicit representation in Banach or Hilbert spaces for multi-order operator evolution problems is achieved via vector-indexed or multivariate Mittag–Leffler functions. For orders $\boldsymbol\beta = (\beta_1,...,\beta_m)$, the solution to linear systems can be written, e.g.,

$$U(t) = E_{\boldsymbol\beta,1}(F(A_1)t^{\beta_1},...,F(A_m)t^{\beta_m})\Phi + \int_0^t S(t-\tau,A) D_\tau^{1-\boldsymbol\beta} H(\tau) d\tau,$$

where $E_{\boldsymbol\beta,1}$ is a vector-indexed Mittag–Leffler function and $F(A)$ encodes the inter-component/coupled dynamics (Umarov, 4 Feb 2024).

• In the presence of non-commuting operators or mixed variable coefficients, operator-valued or non-permutable Mittag–Leffler functions are used for direct time-domain representations (Mahmudov et al., 2021).

Existence and Uniqueness:

• For nonlinear systems with multi-order Caputo or Riemann–Liouville derivatives, the existence and uniqueness of solutions utilize Volterra-type integral reformulations and the Bielecki norm/fixed-point methods under local or global Lipschitz conditions on the nonlinear dynamics (Bourdin, 2017).
• In PDE settings (e.g., time-fractional subdiffusion with space-dependent coefficients and multiple orders), analyticity in time and long-time asymptotics are governed by Laplace-transform techniques, and solutions exhibit decay with rate determined by the smallest order (Li et al., 2018).
• For dynamic-order systems, the existence is guaranteed by the regularity and boundedness of the auxiliary variables driving the order, provided suitable conditions on the memory index map and the coupled system (Sun et al., 2011).

3. Numerical Methods, Approximation, and Consistency

Spectral, Collocation, and Hybrid Function Approaches:

• Runge–Kutta-type schemes based on Fractional Hamiltonian Boundary Value Methods (FHBVMs) have been extended to the multi-order case by employing multiple (vector) Jacobi–Piñeiro polynomial bases and block Gaussian quadrature. This allows high (spectral-like) accuracy, with algebraic order matching the chosen stage polynomial degree, and efficient implementation for up to two distinct orders (Brugnano et al., 4 Dec 2025).
• Orthogonal hybrid function operational matrices enable the transformation of multi-order FDEs (including variable coefficient and nonlinear cases) to finite-dimensional algebraic systems. This approach yields machine-precision accuracy for smooth problems and demonstrates robust performance on both linear and nonlinear multi-order test cases (Damarla et al., 2018).

Consistency in Operator Composition:

• Approximation of fractional integrators/differentiators in control and circuit settings must respect algebraic identities:

$$\mathscr{I}^\alpha \mathscr{I}^{1-\alpha} = 1/s, \quad \mathscr{D}^\alpha \mathscr{I}^\alpha = 1, \quad \mathscr{D}^\alpha \mathscr{D}^{1-\alpha} = s$$

which are generically violated by classic approximants such as Oustaloup filters. Piecewise rational models with explicit design of pole–zero structures now allow satisfaction of these identities in multi-order systems, yielding exact cancellation properties in cascadable controllers and improved time- and frequency-domain performance (Wei et al., 2021).

Discretizations for Auxiliary-Driven/Dynamic-Order Models:

• Variable-order L1 schemes, with stepwise updates of the order-dependent weights, offer a direct route for simulating systems with memory order modulated by a coupled dynamical process. These algorithms automatically adapt the memory kernel at each time step, capturing the nonstationary behavior driven by multiscale or auxiliary interactions (Sun et al., 2011).

4. Inverse Problems and Parameter Identification

Uniqueness in inverse problems for multi-order systems emerges from the coupling structure. In diffusive networks (coupled subdiffusion PDEs with different orders), a single observed time trace at one interior point uniquely determines the entire fractional order vector, under “cooperative” coupling and suitable initial data. Discrete minimization approaches, employing Gauss–Newton iterations with finite-difference Jacobians, provide practical and accurate recovery of unknown orders, with stable performance even for noisy and partially observed data. Extensions include higher spatial dimensions, general regularization, and adjoint-based Jacobian computation for high efficiency (Liu, 3 Feb 2025, Li et al., 2018).

5. Optimal Control and Pontryagin Principles in Multi-Order Systems

Optimal control for systems featuring multiple (possibly incommensurate) Caputo fractional orders employs a vectorized Pontryagin maximum principle, where the adjoint (co-state) equations themselves become multi-order fractional differential equations. The Hamiltonian system must be augmented by fractional transversality conditions and solved as a fractional two-point boundary value problem. Solutions to this system showcase the additional complexity in the adjoint evolution and in the stationarity conditions—reflecting the extended memory effects—and recover the classical PMP as a limiting case when all orders tend to unity. Applications include systems with disparate time-memory scaling and nonlocal optimality constraints (Ndairou et al., 2023).

6. Applications and Physical Interpretation

Multi-order and dynamic-order fractional systems have become central in modeling:

• Composite or layered media: Each fractionally differentiated term models a phase or layer with a distinct memory scaling, e.g., in heat conduction or groundwater flow across heterogeneity, bi-order models parameterize both local and nonlocal effects (Atangana, 2016).
• Multi-physics and multi-scale couplings: Dynamic-order models succinctly encode how transport or relaxation exponents are modulated by auxiliary fields (temperature, stress, chemical concentration), rather than introducing ad hoc coupling terms (Sun et al., 2011).
• Electronic components (Fractor circuits): Devices whose impedance exponent is empirically order-dependent (e.g., on temperature) are prototypical dynamic-order systems—analytically and in control realization (Sun et al., 2011, Wei et al., 2021).
• Viscoelastic/rheological modeling: Memory exponents modulated by environmental variables naturally map to multi-order or dynamic-order frameworks, capturing richer transition and creep phenomena (Sun et al., 2011, Atangana, 2016).
• Multi-objective optimization: Recent adaptive-order Caputo gradient descent algorithms leverage the flexibility of per-iteration fractional order selection, generalizing fixed-order methods and proving robust in non-smooth multi-objective settings (Shaw et al., 10 Jul 2025).

7. Open Problems and Directions

Several technical and application-driven directions remain open:

• Efficient and stable high-order discretizations for arbitrary numbers of orders, especially in stiff, nonlinear, or high-dimensional systems.
• Extension of inverse problem theory to regimes with partial observation, measurement corruption, or incomplete coupling.
• Parameter identification for complex multi-order kernels, especially in variable- or auxiliary-driven order scenarios.
• Universal frameworks for operator theory and state-transition matrices in multi-order systems beyond two or three components (Umarov, 4 Feb 2024, Bourdin, 2017).
• Rigorous justification and error analysis for numerical and model reduction schemes specific to multi-order and dynamic-order PDEs.
• Broader application in biological, neural, economic, and engineered systems where heterogeneous memory effects and inter-system modulation are predominant.

Fractional multi-order problems thus constitute a fundamental and rapidly evolving area in fractional calculus, unifying advanced operator theory, numerical analysis, inverse problem methodology, and modern control, with direct connections to the modeling demands of contemporary multi-scale, multi-physics systems.