Atangana-Baleanu-Caputo Operator

• The Atangana-Baleanu-Caputo operator is a fractional derivative with a nonlocal, nonsingular Mittag–Leffler memory kernel, providing smooth interpolation between power-law and exponential decays.
• It improves on classical Caputo and Riemann–Liouville formulations by eliminating initial singularities and reducing high-frequency amplification in various applications.
• The operator finds practical use in signal processing, viscoelasticity, and anomalous diffusion, offering enhanced numerical stability and well-defined memory effects.

The Atangana–Baleanu–Caputo (ABC) operator is a fractional derivative featuring a nonlocal, nonsingular memory kernel constructed from the Mittag–Leffler function. It was introduced to overcome certain limitations of classical Caputo and Riemann–Liouville fractional calculus, particularly the strong singularity at the initial point and overly aggressive high-frequency amplification in applications such as signal analysis, viscoelasticity, and anomalous diffusion. The ABC derivative interpolates between heavy-tailed, power-law and exponentially tempered memory, yielding improved smoothing properties, boundedness, and well-defined operational calculus while retaining genuine fractional dynamics.

1. Definition and Fundamental Representation

Let $0<\alpha<1$ and $f$ be sufficiently smooth on $[0,T]$. The Atangana–Baleanu–Caputo fractional derivative of order $\alpha$ is defined as a Caputo-type convolution: $${}^{ABC}D_t^\alpha f(t) = \frac{B(\alpha)}{1-\alpha} \int_0^t E_\alpha\left(-\frac{\alpha}{1-\alpha}(t-\tau)^\alpha\right) f'(\tau) \, d\tau,$$ where $$E_\alpha(z) = \sum_{k=0}^\infty \frac{z^k}{\Gamma(\alpha k + 1)}$$ is the one-parameter Mittag–Leffler function, and $B(\alpha)$ is a normalization constant that guarantees ${}^{ABC}D_t^\alpha f \to f'$ as $\alpha \to 1^-$ and $B(0)=B(1)=1$ (Demir et al., 15 Dec 2025, Giusti, 2017, Demir et al., 15 Dec 2025).

In the Laplace domain, for $F(s)=\mathcal{L}\{f\}(s)$,

$$\mathcal{L}\{ {}^{ABC}D_t^\alpha f(t)\}(s) = \Phi_\alpha(s) \left[F(s) - \frac{f(0)}{s}\right]$$

with

$$\Phi_\alpha(s) = B(\alpha) \frac{s^\alpha}{1-\alpha + \alpha s^\alpha}.$$

Unlike the singular Caputo kernel (power-law $(t-\tau)^{-\alpha}$), the ABC kernel decays via the entire, nonsingular Mittag–Leffler function, producing a regular memory profile at short times and a fractional power-law at long times (Giusti, 2017).

2. Analytical Structure and Properties

2.1 Linearity and Limiting Cases

• Linearity: ${}^{ABC}D_t^\alpha$ is linear.
• Consistency: As $\alpha \to 1$, the operator converges to the classical derivative; as $\alpha \to 0$, it reduces to the identity (Demir et al., 15 Dec 2025).

2.2 Memory Effect and Frequency Response

The ABC derivative encodes full history dependence:

• The present rate of change depends on $f'(\tau)$ over all $\tau\le t$ with weights governed by the Mittag–Leffler kernel.
• Smaller $\alpha$ yields longer memory (slower kernel decay), while $\alpha \to 1$ recovers near-instantaneous response.
• Frequency response: at low frequencies, $\Phi_\alpha(s) \sim s^\alpha$ (classical fractional scaling); at high frequencies, the gain saturates, enhancing numerical stability and suppressing over-amplification (Demir et al., 15 Dec 2025).

2.3 Comparison with Standard Fractional Operators

• Riemann–Liouville/Caputo: Singular, algebraic tails with unbounded kernel at $t\to\tau$.
• Caputo–Fabrizio: Exponential kernel, finite memory but lacks algebraic tail.
• Atangana–Baleanu–Caputo: Nonsingular, intermediate memory—interpolating between exponential and algebraic decay (Demir et al., 15 Dec 2025, Douaifia et al., 2020).

2.4 Series and Prabhakar Integral Representation

The ABC derivative is a special case of a Prabhakar (three-parameter Mittag–Leffler) fractional integral: $${}^{ABC}D_{a+}^\alpha f(t) = \frac{B(\alpha)}{1-\alpha} E_{\alpha,1,\omega}^{1}[f'](t),$$ where $E_{\alpha,\beta}^\gamma(z)$ is the Prabhakar function and $\omega = -\alpha/(1-\alpha)$. This results in a convergent series of Riemann–Liouville fractional integrals: $${}^{ABC}D_{a+}^\alpha f(t) = \frac{B(\alpha)}{1-\alpha} \sum_{k=0}^\infty \omega^k J_{a+}^{\alpha k+1}[f'](t).$$ This links the operator structurally to the classical fractional calculus hierarchy (Giusti, 2017).

3. Existence, Uniqueness, and Analytical Results

3.1 Volterra–Type Integral Representation

Any nonlinear fractional differential equation of the form

$${}^{ABC}D_{0+}^\alpha w(t) = f(t, w(t)), \quad w(0)=w_0$$

is equivalent to

$$w(t) = w_0 + \frac{1-\alpha}{B(\alpha)} f(t, w(t)) + \frac{\alpha}{B(\alpha)\Gamma(\alpha)} \int_0^t (t-s)^{\alpha-1} f(s, w(s)) ds.$$

Analyses of local (Peano-type) existence, a priori bounds, and extremality follow using standard fixed-point arguments (Kucche et al., 2020, Demir et al., 15 Dec 2025, Ammi et al., 2021). Uniqueness is ensured under Lipschitz conditions with suitable radius and step-size restrictions inherited from the Caputo-case theory (Owolabi et al., 2021).

3.2 Comparison Principles and Extreme-Point Results

• Comparison results, including a Gronwall-type inequality and maximum/minimum principles, hold due to the positivity and regularity of the Mittag–Leffler kernel.
• Extremal point estimates ensure sign preservation of the ABC derivative at locations of maximality/minimality (Kucche et al., 2020).

4. Discrete and Computational Realizations

4.1 Discrete ABC and Difference–Sum Operators

Discrete analogues are constructed via rising factorials and discrete Mittag–Leffler kernels,

$$\mathrm{ABC}^\alpha_{a} f(t) = B(\alpha)(1-\alpha) \sum_{s=a+1}^t f(s) E_{\alpha}(1-\alpha; t-\rho(s)).$$

Iterated versions with a secondary parameter satisfy a semigroup property via the binomial theorem: $$AB\nabla_{a}^{(-\alpha, p)}(AB\nabla_{a}^{(-\alpha, q)}f)(t) = AB\nabla_{a}^{(-\alpha, p+q)}f(t)$$ with simple Laplace-domain representations, ensuring analytical tractability (Abdeljawad et al., 2019).

4.2 Predictor–Corrector and Adams–Bashforth–Moulton Schemes

Numerical integration commonly relies on predictor–corrector constructions—either Newton-interpolated or Adams–Bashforth–Moulton type schemes—where the ABC memory kernel appears in discrete convolutional sums with efficiently precomputed weights derived from the discretized Mittag–Leffler kernel (Djida et al., 2016, Douaifia et al., 2020, Owolabi et al., 2021, Ammi et al., 2021, Ryehan, 2023, Demir et al., 15 Dec 2025).

4.3 Error, Stability, and Implementation

• The error of discrete ABC-integral schemes is typically $O(\Delta t)$ under regularity assumptions.
• No CFL-type restriction is required for temporal discretization.
• The PECE Adams–Moulton strategy, together with standard step-size selection, ensures convergence and $L^\infty$ boundedness of the solution (Djida et al., 2016, Ammi et al., 2021).

5. Applications and Comparative Behavior

5.1 Signal Processing and Inverse Problems

The ABC derivative is particularly effective in scenarios where preservation of memory and suppression of high-frequency noise are critical. In fractional wavelet compression for astronomical time-series, ABC-regularized flows at the coefficient level enable gradient-regularization with memory, outperforming classical wavelet-thresholding in preserving weak transients and suppressing artifacts (Demir et al., 15 Dec 2025).

5.2 Viscoelasticity and Seismic Wave Modeling

In viscoelastic wave equations, replacing conventional derivatives with the ABC operator introduces a physically realistic, non-singular hereditary term. Simulations replacing Caputo models by ABC show pronounced non-exponential energy decay and enhanced long-memory damping and dispersion—capabilities essential for modeling anomalously attenuated seismic waves (Demir et al., 15 Dec 2025).

5.3 Fractional Evolution Equations in Banach Spaces

The ABC operator supports formulations of fractional Cauchy problems with almost sectorial operators, admitting well-posedness, smoothing estimates, and explicit resolvent representations in the Laplace domain: $$D_t^{ABC,\alpha} u(t) = Au(t) + f(t), \quad u(0) = u_0,$$ where the resolvent family $S_{ABC}(t)$ yields temporal smoothing $\|A^\gamma S_{ABC}(t)\| \leq Ct^{-\alpha\gamma}$, providing instant higher regularity of solutions for $t>0$ (Wakrim, 6 Jan 2026).

5.4 Anomalous Diffusion and Limitations

When introduced into Fokker–Planck and generalized Langevin equations, the ABC operator leads to non-physical features—such as finite initial spread and divergence of moments in cases where power-law subdiffusion is expected—rendering it unsuitable for certain anomalous transport applications requiring traditional Markov or Caputo-type behavior (Wei et al., 2023).

6. Structural and Theoretical Interpretation

6.1 ABC as a Special Prabhakar Integral

The ABC operator is shown to be a special case of the Prabhakar fractional integral, with explicit structural correspondence to an infinite mixture of Riemann–Liouville integrals of variable order. As a result, any ODE or PDE involving ABC can be recast entirely in terms of traditional Caputo or RL operators plus structured memory terms (Giusti, 2017).

6.2 Non-Uniqueness and Redundancy in Fractional Modeling

While the ABC operator provides analytical smoothness and regularized memory, it does not generate dynamical behavior or phenomenology outside the Prabhakar fractional calculus framework. In particular, substituting ABC for Caputo in viscoelastic constitutive equations yields no fundamentally new relaxation or damping laws; all responses can already be captured by the Scott–Blair (fractional Maxwell) model with rescaled parameters (Giusti, 2017).

Operator	Kernel	Singularity at $t=\tau$	Large-$t$ Memory
Caputo	$(t-\tau)^{-\alpha}$	Yes	Power-law
Caputo–Fabrizio	$\exp(-c(t-\tau))$	No	Exponential
Atangana–Baleanu	$E_\alpha(-c(t-\tau)^\alpha)$	No	Power/Mittag–Leffler

7. Conclusion and Scope of Application

The Atangana–Baleanu–Caputo operator provides an analytically robust, numerically stable, and physically interpretable tool for modeling fractional dynamics with nonsingular memory. Its kernel enables a flexible interpolation between exponential and heavy-tailed memory, yielding improved noise suppression and regularization in areas like signal decomposition and long-memory PDEs. For standard applications in viscoelasticity and diffusion, it is structurally equivalent to classical Prabhakar and Caputo-type approaches, adding no fundamentally new behaviors but offering operational convenience and improved numerical conditioning (Demir et al., 15 Dec 2025, Giusti, 2017, Wakrim, 6 Jan 2026). Care is warranted when deploying ABC operators in physically derived stochastic models, as the introduction of finite variance at $t=0$ or divergent moments may be inconsistent with expected physical constraints (Wei et al., 2023).