Generalized Caputo Operators

• Generalized Caputo operators are extensions of the classical Caputo derivative that use variable kernels, measures, and time-varying coefficients to model memory effects and anomalous processes.
• They are applied in fractional calculus, stochastic processes, and variational analysis, providing flexible tools for modeling nonlocality and subdiffusive behavior.
• Key properties such as linearity, vanishing on constants, and inversion via fractional integrals enable robust analytical and numerical treatments in both continuous and discrete settings.

A generalized Caputo operator is any extension or modification of the classical Caputo fractional derivative, often designed to accommodate additional structural features (e.g., variable order, non-power-law kernels, time-varying coefficients, multi-sided memory, or integration over generalized measures) while retaining the defining Caputo property: acting on constants yields zero. These operators appear throughout modern fractional calculus, stochastic processes, and variational analysis and are key tools in modeling memory, non-locality, and anomalous transport phenomena.

1. Structural Definition and Core Variants

The classical Caputo derivative of order $0<\alpha<1$ at the left endpoint $a$ is given by

$${}^C D_{a+}^\alpha f(t) = \frac{1}{\Gamma(1-\alpha)}\int_a^t (t-s)^{-\alpha} f'(s)\, ds.$$

Generalized Caputo operators systematically replace the kernel $(t-s)^{-\alpha}$ or the integration structure with more general constructs. Examples include:

• Generalized Caputo via Lévy measure: For a Borel measure $\nu$ on $(0,\infty)$,

$$D_{a+*}^\nu f(x) = - \int_0^{x-a}[f(x-y) - f(x)]\, \nu(dy) - [f(a)-f(x)]\int_{x-a}^\infty \nu(dy),$$

with $f$ constant on $(-\infty, a]$. Setting $\nu(dy)= y^{-1-\beta}/\Gamma(-\beta)\,dy$ recovers the standard Caputo derivative (Kolokoltsov, 2017).

• Combined Caputo derivatives: For $0<\alpha,\beta<1$, $\gamma \in [0,1]$,

$${}^C D_\gamma^{\alpha,\beta}f(t) = \gamma\, {}^C D_{a+}^\alpha f(t) + (1-\gamma)\, {}^C D_{b-}^\beta f(t),$$

interpolating between past and future memory (Malinowska et al., 2010, Malinowska et al., 2011).

• Caputo–Katugampola (generalized Caputo–Hadamard):

$${}^C D_{a+}^{\alpha,\rho} x(t) = \frac{\rho^{1-\alpha}}{\Gamma(1-\alpha)} t^{1-\rho}\frac{d}{dt} \int_a^t \tau^{\rho-1}(t^\rho - \tau^\rho)^{-\alpha} [x(\tau) - x(a)]\, d\tau,$$

with $\rho>0$. Recovers the Caputo case for $\rho=1$, Caputo–Hadamard as $\rho\to 0^+$ (Almeida, 2016, Almeida, 2017, Almeida et al., 2016).

• Generalized Caputo with arbitrary kernels:

$${}^C D_a^{\alpha,\psi} f(t) = I_a^{1-\alpha, \psi} (f')(t),$$

where $I_a^{\alpha, \psi} f$ is a generalized fractional integral with kernel $\psi_\alpha(t,s)$ (Odzijewicz et al., 2012).

• Discrete, nabla, or delta Caputo operators: Replacing differential/integral operators with their discrete/delta analogs for time scale and lattice settings (Pachpatte, 2019, Mohammed et al., 2022).
• Caputo operators with non-power-law kernels: Caputo–Fabrizio (exponential), Atangana–Baleanu (Mittag-Leffler), and others (Wei et al., 2023, Kataria et al., 2014, Fahad et al., 2019).

A wide range of new physical, stochastic, and analytical behaviors are accessible by suitable selection of the underlying measure, kernel, or combination parameters.

2. Analytical Properties and Operator Calculus

Generalized Caputo operators typically preserve:

• Linearity: $$D^* (\lambda_1 f_1 + \lambda_2 f_2) = \lambda_1 D^*f_1 + \lambda_2 D^*f_2$$ for all $f_1,f_2$.
• Vanishing on constants: $D^*[1]=0$, ensuring compatibility with classical derivative when $\alpha\to 1$.
• Inversion formulas: There exists a generalized fractional integral $I^*$ such that $I^* D^*[f](t) = f(t) - f(a)$ under regularity assumptions—mirroring the classical Caputo calculus (Almeida, 2016, Almeida et al., 2016, Almeida, 2017, Odzijewicz et al., 2012).
• Integration by parts: Generalizations appear, involving right-sided or “dual” Caputo operators and corresponding fractional integrals (Malinowska et al., 2010, Malinowska et al., 2011, Odzijewicz et al., 2012).

The kernel class defines the spectral properties and time-scaling of the induced nonlocality. For instance, in the self-similar multiplicative-convolution approach, there is a bijection with Bernstein functions, providing a flexible spectral calculus and generalized Mittag-Leffler functions as eigenfunctions (Patie et al., 2019).

3. Stochastic and Probabilistic Representations

Generalized Caputo operators commonly act as generators of killed, stopped, or time-changed Feller processes—typically subordinators or jump processes with a specified jump measure. For instance, for a subordinator with Lévy measure $\nu$, the generator is

$$-A^{(\nu)}f(x) = \int_0^\infty [f(x+y) - f(x)]\nu(dy),$$

with the generalized Caputo derivative associated to the process killed or reflected at the boundary. The operator characterizes exit time problems and enables stochastic representations of solutions, such as

$$u(x) = \mathbb{E}\Bigl[\int_0^{\tau_{(a,b)}(x)} e^{-\lambda t} g(X_x(t))\, dt\Bigr]$$

for a generalized Caputo ODE with source $g$ (Hernández-Hernández et al., 2015, Hernández-Hernández et al., 2017, Kolokoltsov, 2017).

This stochastic interpretation underlies the physically-relevant subdiffusion, continuous-time random walks, and models of anomalous relaxation.

4. Special Cases, Reductions, and Kernel Choices

Many generalized Caputo operators encompass, as limiting or special cases:

• Classical Caputo ($\alpha$ fixed, power-law kernel)
• Caputo–Hadamard (logarithmic kernel as $\rho \to 0$)
• Right-sided and combined Caputo (two-sided memory: convex combinations of left/right derivatives)
• Caputo–Fabrizio/Atangana–Baleanu: exponential or Mittag-Leffler kernel, modeling finite or power-law-tailed memory
• Fractional-difference operators: discrete time/fractional calculus on time scales (Pachpatte, 2019)

This inclusion preserves the universality and modeling power of the generalized Caputo approach.

5. Applications and Variational Principles

Generalized Caputo operators have expansive utility:

• Evolution equations: Abstract Cauchy problems with $D^*[u]=A(u)+g$
• Variational calculus: Fractional Euler–Lagrange equations involving generalized Caputo derivatives, natural boundary/transversality conditions, isoperimetric constraints, and Herglotz-type problems (Malinowska et al., 2010, Malinowska et al., 2011, Almeida, 2016, Odzijewicz et al., 2012)
• Operator-valued evolutions: Solutions via generalized Mittag-Leffler functions and Feynman-Kac representations (Kolokoltsov, 2017)
• Special function theory: Action on $K$-Wright, Bessel–Maitland, Wright, and Mittag-Leffler functions, including structural shifts under Caputo–type Marichev–Saigo–Maeda operators (Kataria et al., 2014)
• Anomalous diffusions: Fokker–Planck and generalized Langevin equations incorporating Caputo variants; physical veracity of the model depends crucially on kernel selection (e.g., only power-law Caputo gives correct subdiffusion and initialization) (Wei et al., 2023)

A key structural insight is that the solution theory—existence, uniqueness, well-posedness—extends to many generalized Caputo operators, under suitable regularity and contractiveness assumptions, via fixed-point, integral equation, and stochastic approaches.

6. Discrete, Time-Scale, and Nonstandard Domains

The theory extends to:

• Dynamic equations on time scales: Caputo fractional-delta operators naturally interpolate between continuous, discrete, and $q$-difference settings (Pachpatte, 2019, Mohammed et al., 2022)
• Generalized Dirac, Laplacian, and wave operators: Construction of space-time fractional operators through Caputo and generalized kernels with respect to another function or coordinate transform (Restrepo et al., 2021)

This brings fractional calculus into discrete optimization, network dynamics, quantum calculus, and other nonstandard domains.

7. Summary Table: Representative Generalized Caputo Operators

Construction	Definition/Kernel	Reference(s)
Measure-based (Lévy)	$\nu(dy)$, fully general	(Kolokoltsov, 2017)
Combined left/right Caputo	$\gamma, \alpha, \beta$ mixture	(Malinowska et al., 2010)
Katugampola–Caputo	$(t^\rho-\tau^\rho)^{-\alpha}$ kernel	(Almeida, 2016)
MSM (Appell $F_3$ kernel)	Appell function-based convolution	(Kataria et al., 2014)
Arbitrary kernel (Odzijewicz)	$\psi_\alpha(t,s)$	(Odzijewicz et al., 2012)
Substantial (exponential)	$$(t^\rho-s^\rho)^{\alpha-1} e^{-\sigma (t^\rho-s^\rho)}$$	(Fahad et al., 2019)
Caputo-delta (time scales)	$h_\alpha(t, s)$ parametric kernel	(Pachpatte, 2019)
Self-similar (Bernstein function)	kernel from $m(r)$; operator $D_\Phi$	(Patie et al., 2019)
Caputo-Fabrizio, Atangana-Baleanu	Exponential/Mittag-Leffler kernel	(Wei et al., 2023)

The selection of the kernel, measure, or combination parameters directly dictates the operator’s functional, spectral, and stochastic properties, with physical, modeling, and analytical consequences for the resulting dynamical systems.