Distributed Order Caputo Derivative

• Distributed order Caputo derivative is a weighted integral of conventional Caputo derivatives over a continuum of orders, modeling complex memory effects.
• It enables the analysis of multi-scale phenomena in anomalous diffusion and control systems by employing convolution kernels and rigorous energy estimates.
• The framework ensures existence, regularity, and decay properties in both PDE and ODE models, unifying discrete and continuous fractional operators.

A distributed order Caputo derivative is a linear operator acting on suitable function spaces, defined as a weighted integral over a continuum of fractional Caputo derivatives of variable orders, with the weight function—typically denoted μ—assigning the distribution of orders. This operator generalizes the classical Caputo derivative (of fixed order) and enables the modeling of complex, multi-scale memory effects in evolutionary equations. Key applications arise in anomalous diffusion, fractional variational calculus, and control theory. The mathematical theory necessitates careful treatment of existence, regularity, and decay properties for associated PDEs and ODEs, with the behavior of μ near the endpoints—especially near zero—governing the qualitative dynamics and decay rates of solutions.

1. Formal Definition and Analytical Structure

Let μ : [0,1]→[0,∞) be a nonnegative weight function in $L^1(0,1)$, not identically zero. For $u\in AC[0,T]$ (absolutely continuous), the classical Caputo derivative of order $\alpha\in(0,1)$ is

$${}^C D_t^\alpha u(t) = \frac{1}{\Gamma(1-\alpha)} \int_0^t (t-s)^{-\alpha} u'(s)\, ds,$$

with ${}^C D_t^0 u = u-u(0)$ and ${}^C D_t^1 u = u'(t)$. The distributed order Caputo derivative is then

$$D_t^{(\mu)} u(t) := \int_0^1 \mu(\alpha)\, {}^C D_t^\alpha u(t)\, d\alpha.$$

For each fixed $t$, the operator $(\alpha\mapsto {}^C D_t^\alpha u(t))$ is $L^1$-integrable in $u\in AC[0,T]$0 provided $u\in AC[0,T]$1 and $u\in AC[0,T]$2, so $u\in AC[0,T]$3 exists almost everywhere. This construction encompasses, as special cases, the single-order Caputo derivative (taking μ as a Dirac measure) and the uniform “averaged” Caputo derivative (taking μ constant) (Kubica et al., 2017, Ndairou et al., 2020).

Linearity holds: $u\in AC[0,T]$4 for arbitrary scalars $u\in AC[0,T]$5 (Ndairou et al., 2020). The operator’s action can be rewritten as convolution with a kernel: $u\in AC[0,T]$6 which is critical for both analysis and inversion (Kubica et al., 2017).

2. Functional Framework and Mapping Properties

For problems on a spatial domain $u\in AC[0,T]$7 with $u\in AC[0,T]$8, and $u\in AC[0,T]$9, one considers $\alpha\in(0,1)$0 defined on $\alpha\in(0,1)$1. The function spaces for well-posedness are dictated by the regularity requirements of the distributed order derivative:

• Weak theory: $\alpha\in(0,1)$2, and for almost every $\alpha\in(0,1)$3, $\alpha\in(0,1)$4, ensuring $\alpha\in(0,1)$5 (Kubica et al., 2017).
• Strong/regular theory: $\alpha\in(0,1)$6, $\alpha\in(0,1)$7.

If $\alpha\in(0,1)$8 (scalar or Hilbert-space-valued), then $\alpha\in(0,1)$9 (Kubica et al., 2017). Inversion properties: There exists a right-inverse $${}^C D_t^\alpha u(t) = \frac{1}{\Gamma(1-\alpha)} \int_0^t (t-s)^{-\alpha} u'(s)\, ds,$$0 (a convolution with a kernel in $${}^C D_t^\alpha u(t) = \frac{1}{\Gamma(1-\alpha)} \int_0^t (t-s)^{-\alpha} u'(s)\, ds,$$1) such that $${}^C D_t^\alpha u(t) = \frac{1}{\Gamma(1-\alpha)} \int_0^t (t-s)^{-\alpha} u'(s)\, ds,$$2 for $${}^C D_t^\alpha u(t) = \frac{1}{\Gamma(1-\alpha)} \int_0^t (t-s)^{-\alpha} u'(s)\, ds,$$3, and $${}^C D_t^\alpha u(t) = \frac{1}{\Gamma(1-\alpha)} \int_0^t (t-s)^{-\alpha} u'(s)\, ds,$$4 for $${}^C D_t^\alpha u(t) = \frac{1}{\Gamma(1-\alpha)} \int_0^t (t-s)^{-\alpha} u'(s)\, ds,$$5 (Kubica et al., 2017).

3. Model Equations and Existence Theory

Consider the initial–boundary-value problem: $${}^C D_t^\alpha u(t) = \frac{1}{\Gamma(1-\alpha)} \int_0^t (t-s)^{-\alpha} u'(s)\, ds,$$6 with $${}^C D_t^\alpha u(t) = \frac{1}{\Gamma(1-\alpha)} \int_0^t (t-s)^{-\alpha} u'(s)\, ds,$$7, $${}^C D_t^\alpha u(t) = \frac{1}{\Gamma(1-\alpha)} \int_0^t (t-s)^{-\alpha} u'(s)\, ds,$$8, and $${}^C D_t^\alpha u(t) = \frac{1}{\Gamma(1-\alpha)} \int_0^t (t-s)^{-\alpha} u'(s)\, ds,$$9 a second-order uniformly elliptic operator in divergence form with lower-order terms (Kubica et al., 2017). Under the minimal assumption ${}^C D_t^0 u = u-u(0)$0, ${}^C D_t^0 u = u-u(0)$1, ${}^C D_t^0 u = u-u(0)$2, and suitable integrability for ${}^C D_t^0 u = u-u(0)$3, the following hold:

• Weak solutions: Existence and uniqueness for ${}^C D_t^0 u = u-u(0)$4, ${}^C D_t^0 u = u-u(0)$5. The solution ${}^C D_t^0 u = u-u(0)$6 has an a priori energy estimate involving the ${}^C D_t^0 u = u-u(0)$7 norms of generalized Abel-type integrals of ${}^C D_t^0 u = u-u(0)$8 and of ${}^C D_t^0 u = u-u(0)$9 in ${}^C D_t^1 u = u'(t)$0.
• Regular solutions: With enhanced regularity of data (${}^C D_t^1 u = u'(t)$1, ${}^C D_t^1 u = u'(t)$2, ${}^C D_t^1 u = u'(t)$3), uniqueness and a regularity result in ${}^C D_t^1 u = u'(t)$4 is obtained, including a global-in-time ${}^C D_t^1 u = u'(t)$5-norm estimate on ${}^C D_t^1 u = u'(t)$6, its spatial derivatives, and distributed-order time-increments.

Continuity at ${}^C D_t^1 u = u'(t)$7 is characterized: if ${}^C D_t^1 u = u'(t)$8, then ${}^C D_t^1 u = u'(t)$9 and $$D_t^{(\mu)} u(t) := \int_0^1 \mu(\alpha)\, {}^C D_t^\alpha u(t)\, d\alpha.$$0 (Kubica et al., 2017).

4. Decay Properties and Influence of the Weight Function

The asymptotic decay behavior for the distributed order Caputo diffusion problem is governed by the detailed behavior of the weight function μ near zero. Consider the scalar ODE

$$D_t^{(\mu)} u(t) := \int_0^1 \mu(\alpha)\, {}^C D_t^\alpha u(t)\, d\alpha.$$1

with $$D_t^{(\mu)} u(t) := \int_0^1 \mu(\alpha)\, {}^C D_t^\alpha u(t)\, d\alpha.$$2. Then for large $$D_t^{(\mu)} u(t) := \int_0^1 \mu(\alpha)\, {}^C D_t^\alpha u(t)\, d\alpha.$$3,

$$D_t^{(\mu)} u(t) := \int_0^1 \mu(\alpha)\, {}^C D_t^\alpha u(t)\, d\alpha.$$4

Stronger, explicit decay rates follow from the structure of μ:

• Power law near zero: If $$D_t^{(\mu)} u(t) := \int_0^1 \mu(\alpha)\, {}^C D_t^\alpha u(t)\, d\alpha.$$5 for $$D_t^{(\mu)} u(t) := \int_0^1 \mu(\alpha)\, {}^C D_t^\alpha u(t)\, d\alpha.$$6 small, $$D_t^{(\mu)} u(t) := \int_0^1 \mu(\alpha)\, {}^C D_t^\alpha u(t)\, d\alpha.$$7.
• Gap at zero: If $$D_t^{(\mu)} u(t) := \int_0^1 \mu(\alpha)\, {}^C D_t^\alpha u(t)\, d\alpha.$$8, $$D_t^{(\mu)} u(t) := \int_0^1 \mu(\alpha)\, {}^C D_t^\alpha u(t)\, d\alpha.$$9 for $t$0.
• Uniform weight: For μ constant, “ultraslow” logarithmic decay, $t$1, arises.

For the associated parabolic PDE, the $t$2-norm $t$3 can be bounded above by $t$4 for the scalar problem, so all decay results transfer to spatially extended diffusive settings (Kubica et al., 2017).

5. Special Cases and Connections to Other Fractional Operators

Discrete distribution: If μ is a finite sum of Dirac measures,

$t$5

then $t$6, recovering the multi-term Caputo derivative (Saxena et al., 2011, Malinowska et al., 2010).

Single order: Taking $t$7 yields the classical Caputo derivative $t$8.

Uniform distribution: For μ constant on $t$9, $(\alpha\mapsto {}^C D_t^\alpha u(t))$0 is the average over all orders between 0 and 1, a model for “ultraslow diffusion” (Kubica et al., 2017).

Comparison with Riemann–Liouville: The distributed order Caputo and Riemann–Liouville derivatives differ by an initial value term. Explicitly: $(\alpha\mapsto {}^C D_t^\alpha u(t))$1 where $(\alpha\mapsto {}^C D_t^\alpha u(t))$2 denotes the distributed-order Riemann–Liouville derivative (Ndairou et al., 2020).

6. Analytical Techniques and Solution Strategies

Existence and regularity proofs rely on Galerkin approximations using the eigenfunction basis of $(\alpha\mapsto {}^C D_t^\alpha u(t))$3 on $(\alpha\mapsto {}^C D_t^\alpha u(t))$4 with zero Dirichlet boundary conditions. Time-smoothing and mollification of the coefficients yield finite-dimensional Volterra-type systems involving scalar distributed order Caputo derivatives. A priori energy estimates follow from a fractional energy identity and generalized Grönwall-type inequalities adapted to the distributed-order context.

For the inversion, the kernel $(\alpha\mapsto {}^C D_t^\alpha u(t))$5 defined via convolution with the right-inverse $(\alpha\mapsto {}^C D_t^\alpha u(t))$6 is constructed to satisfy $(\alpha\mapsto {}^C D_t^\alpha u(t))$7 (in $(\alpha\mapsto {}^C D_t^\alpha u(t))$8), permitting explicit characterization of solution regularity and continuity at initial time (Kubica et al., 2017, Kubica et al., 2017).

Laplace transform and Bromwich contour inversion play a central role in analyzing decay for both ODE and PDE settings, with fine asymptotics traced to the singularity structure of $(\alpha\mapsto {}^C D_t^\alpha u(t))$9 as $L^1$0 (Kubica et al., 2017).

7. Applications and Related Models

Fractional diffusion: The distributed order Caputo derivative provides a rigorous framework for modeling anomalous diffusive behavior with memory distributed over a range of temporal scales, encompassing ultraslow and super-slow diffusion regimes (Kubica et al., 2017, Kubica et al., 2017).

Reaction–diffusion systems: In models where the time derivative is replaced by a linear combination (or more generally, an integral) of Caputo derivatives of different orders, one recovers distributed order kinetics. Solutions often expressible in terms of special functions (e.g., Fox H-functions) and admit a unified approach covering telegraph, fractional diffusion, and space-fractional cases (Saxena et al., 2011).

Fractional variational principles and control: Distributed order Caputo operators underlie generalizations of variational calculus and optimal control. Euler–Lagrange equations and necessary/sufficient conditions incorporate distributed order dynamics, with the distributed order derivative recast as a continuum or discrete average (with δ-mass weights) over classical Caputo orders (Ndairou et al., 2020, Malinowska et al., 2010).

Specialization Type	μ(α) Structure	Operator Interpretation
Single Caputo	Dirac δ	Fixed-order Caputo
Multi-term (discrete)	Finite sum δ	Sum of fixed-order Caputo
Uniformly distributed	μ constant	Averaged-order Caputo
General distributed-order	μ∈L¹(0,1), μ≥0	Superposition (continuum of orders)

The distributed order Caputo derivative thus constitutes a natural extension of the fractional calculus toolkit, unifying discrete, multi-term, and continuous distributed-order phenomena within a technically rigorous framework (Kubica et al., 2017, Ndairou et al., 2020, Kubica et al., 2017, Saxena et al., 2011).