Sequential Caputo Derivatives in Nonlocal Operators

• Sequential Caputo derivatives are compositions of fractional operators that do not follow the index-addition rule, acting as distinct nonlocal operators.
• They require extra initial traces and boundary information, affecting analysis and solution strategies in PDE and variational contexts.
• Numerical methods use graded time discretizations and Fourier techniques with bivariate Mittag-Leffler functions for accurate resolution.

Sequential Caputo derivatives are compositions of Caputo fractional derivatives, typically of the form

${}^{C}D_t^\beta\bigl({}^{C}D_t^\alpha u\bigr),$

and are studied as genuine iterated nonlocal operators rather than as a single Caputo derivative of order $\alpha+\beta$. In the recent PDE literature, this distinction is explicit: for $0<\alpha,\beta<1$, the operator $D_t^\beta(D_t^\alpha f)$ is treated as a sequential composition with its own initial data, resolvent structure, and kernel calculus, and it is stated that $D_t^\beta(D_t^\alpha f)\neq D_t^{\alpha+\beta}f$ (Yusuf et al., 23 Apr 2026). Generalized sequential problems of the same type also appear for $\psi$-Caputo derivatives, where the composition is analyzed through Green functions, nonlocal boundary conditions, and weighted integral formulations rather than through a semigroup law (Dien, 2021).

1. Definition and conceptual status

For $0<\gamma<1$, the Caputo derivative is written in the form

$D_t^\gamma f(t)=I_t^{1-\gamma}\frac{d}{dt}f(t),$

with

$$I_t^\sigma f(t)=\frac{1}{\Gamma(\sigma)}\int_0^t (t-\xi)^{\sigma-1}f(\xi)\,d\xi.$$

The sequential Caputo derivative is then the composition

$$D_t^\beta\!\left(D_t^\alpha f(t)\right) = \left(I_t^{1-\beta}\frac{d}{dt}\right)\left(I_t^{1-\alpha}\frac{d}{dt}f(t)\right),$$

which is the explicit definition used for the model problem

$\alpha+\beta$0

on $\alpha+\beta$1 (Yusuf et al., 23 Apr 2026).

A defining feature of the subject is that sequential Caputo differentiation is not identified with an index-addition rule. The statement

$\alpha+\beta$2

is made explicitly in the PDE analysis of (Yusuf et al., 23 Apr 2026). The generalized $\alpha+\beta$3-Caputo theory proceeds in the same spirit: it studies

$\alpha+\beta$4

as a composition, while using only the semigroup property for the associated fractional integrals,

$\alpha+\beta$5

and not for the $\alpha+\beta$6-Caputo derivatives themselves (Dien, 2021).

This distinction separates sequential Caputo derivatives from two different but nearby notions. First, a single higher-order Caputo derivative with order $\alpha+\beta$7 is still one operator defined through the ceiling $\alpha+\beta$8, not a repeated Caputo differentiation; this is the standpoint of the shifted Gegenbauer pseudospectral work on arbitrary positive orders (Elgindy, 29 Jan 2025). Second, composition of a fractional integral with a Caputo derivative,

$\alpha+\beta$9

is a different problem from sequential Caputo differentiation; the discrete analogue studied for the L1 scheme does not address $0<\alpha,\beta<1$0 (Płociniczak, 2021).

2. Initial data, domains, and boundary terms

Sequential Caputo derivatives require richer boundary-initial data than single-order fractional models. For the boundary-initial value problem

$0<\alpha,\beta<1$1

the imposed data are

$0<\alpha,\beta<1$2

so the fractional initial trace $0<\alpha,\beta<1$3 is part of the problem specification (Yusuf et al., 23 Apr 2026). This reflects the intermediate state created by the first Caputo operation.

From the operator-theoretic viewpoint, the main issue is domain compatibility. In the Sobolev-space realization of the Caputo derivative for $0<\alpha,\beta<1$4,

$0<\alpha,\beta<1$5

where $0<\alpha,\beta<1$6 is the Riemann–Liouville fractional integral (Gorenflo et al., 2014). The same framework implies that a sequential derivative $0<\alpha,\beta<1$7 is meaningful only if

$0<\alpha,\beta<1$8

That paper does not prove a composition theorem, but it makes precise that sequential Caputo differentiation is прежде всего a domain question (Gorenflo et al., 2014).

Boundary regularization is equally fundamental. In the probabilistic-generator treatment of Caputo and Riemann–Liouville operators, the Caputo derivative is presented as an RL derivative corrected by boundary terms. For $0<\alpha,\beta<1$9,

$D_t^\beta(D_t^\alpha f)$0

and for $D_t^\beta(D_t^\alpha f)$1,

$D_t^\beta(D_t^\alpha f)$2

in the form stated there (Kolokoltsov, 2015). This suggests that any sequential Caputo calculus must track the boundary traces produced after the first differentiation; a naive semigroup formula would ignore precisely the terms that distinguish Caputo from RL behavior.

A closely related, but explicitly marked, inference comes from the generalized convolution-group approach to Caputo derivatives. That work extends Caputo derivatives to locally integrable functions and emphasizes singularities at $D_t^\beta(D_t^\alpha f)$3 so that a group property holds for the underlying fractional calculus; a plausible implication is that the failure of a naive sequential Caputo law is naturally represented by singular terms supported at the initial time (Li et al., 2016).

3. Boundary value problems and exact solution theory

The most direct analytical treatment of sequential Caputo derivatives in the supplied literature appears in two settings: a classical time-fractional PDE and a generalized $D_t^\beta(D_t^\alpha f)$4-Caputo boundary value problem.

For the PDE

$D_t^\beta(D_t^\alpha f)$5

with $D_t^\beta(D_t^\alpha f)$6, $D_t^\beta(D_t^\alpha f)$7, $D_t^\beta(D_t^\alpha f)$8, and $D_t^\beta(D_t^\alpha f)$9, the solution is constructed by Fourier sine expansion

$D_t^\beta(D_t^\alpha f)\neq D_t^{\alpha+\beta}f$0

which reduces the problem modewise to

$D_t^\beta(D_t^\alpha f)\neq D_t^{\alpha+\beta}f$1

The Laplace-domain resolvent has denominator

$D_t^\beta(D_t^\alpha f)\neq D_t^{\alpha+\beta}f$2

not $D_t^\beta(D_t^\alpha f)\neq D_t^{\alpha+\beta}f$3, and this two-scale structure leads to an exact representation in terms of the bivariate Mittag-Leffler function $D_t^\beta(D_t^\alpha f)\neq D_t^{\alpha+\beta}f$4: each modal solution is a sum of two homogeneous $D_t^\beta(D_t^\alpha f)\neq D_t^{\alpha+\beta}f$5-terms and one $D_t^\beta(D_t^\alpha f)\neq D_t^{\alpha+\beta}f$6-convolution term (Yusuf et al., 23 Apr 2026).

The same paper formulates integral and Caputo-differentiation identities for $D_t^\beta(D_t^\alpha f)\neq D_t^{\alpha+\beta}f$7, proves an estimate of the form

$D_t^\beta(D_t^\alpha f)\neq D_t^{\alpha+\beta}f$8

under the stated parameter restrictions, and uses these results to justify convergence of the sine series. Under the assumptions

$D_t^\beta(D_t^\alpha f)\neq D_t^{\alpha+\beta}f$9

existence and uniqueness of a regular solution are established (Yusuf et al., 23 Apr 2026).

In the generalized setting, the nonlinear sequential $\psi$0-Caputo boundary value problem

$\psi$1

with

$\psi$2

is converted into the integral equation

$\psi$3

with an explicit piecewise Green function (Dien, 2021). The kernel contains the factors

$\psi$4

which encode both the total order $\psi$5 and the intermediate order $\psi$6. From this representation the paper derives a Lyapunov-type inequality, a lower bound for possible eigenvalues, and existence and uniqueness results via Leray–Schauder and contraction arguments (Dien, 2021).

4. Variational and generalized operator contexts

Sequential Caputo structures also arise indirectly in fractional variational calculus. A foundational obstacle is that functionals depending on Caputo derivatives usually lead, after integration by parts, to Euler–Lagrange equations involving Riemann–Liouville derivatives. For Lagrangians with left and right Caputo derivatives, the fractional Euler–Lagrange equation takes the form

$\psi$7

so the composed structure is mixed Caputo–RL rather than Caputo–Caputo (Almeida et al., 2011).

A different variational approach produces Euler–Lagrange equations involving only Caputo derivatives,

$\psi$8

and for quadratic dependence on the Caputo variable this yields the explicit left-right composition

$\psi$9

as a special case (Lazo et al., 2012). That operator is not the same as the one-sided sequential derivative $0<\gamma<1$0, but it shows that genuine Caputo-after-Caputo structures occur naturally in variational models.

These adjacent theories clarify a recurrent point: the phrase “sequential Caputo derivative” does not name a single universal composition rule. In one-sided initial-value problems it refers to iterated time differentiation such as $0<\gamma<1$1 (Yusuf et al., 23 Apr 2026); in generalized boundary value problems it refers to $0<\gamma<1$2-Caputo compositions (Dien, 2021); in variational settings it often appears as left-right compositions (Lazo et al., 2012).

5. Numerical approximation and discrete composition

The numerical treatment of sequential Caputo derivatives currently proceeds mainly by reformulation rather than by direct discretization of a nested operator. In the PDE study (Yusuf et al., 23 Apr 2026), the auxiliary variable

$0<\gamma<1$3

is introduced, so that the sequential model becomes

$0<\gamma<1$4

Space is discretized by conforming $0<\gamma<1$5 finite elements, time by the L1 approximation on a graded mesh

$0<\gamma<1$6

and the fully discrete system is solved sequentially for the nodal vectors $0<\gamma<1$7 and $0<\gamma<1$8 (Yusuf et al., 23 Apr 2026). For the manufactured solution

$0<\gamma<1$9

with $D_t^\gamma f(t)=I_t^{1-\gamma}\frac{d}{dt}f(t),$0, $D_t^\gamma f(t)=I_t^{1-\gamma}\frac{d}{dt}f(t),$1, $D_t^\gamma f(t)=I_t^{1-\gamma}\frac{d}{dt}f(t),$2, $D_t^\gamma f(t)=I_t^{1-\gamma}\frac{d}{dt}f(t),$3, and $D_t^\gamma f(t)=I_t^{1-\gamma}\frac{d}{dt}f(t),$4, the reported errors are

$D_t^\gamma f(t)=I_t^{1-\gamma}\frac{d}{dt}f(t),$5

and graded meshes are observed to improve the rates relative to uniform meshes (Yusuf et al., 23 Apr 2026).

Two nearby numerical literatures are relevant but should be distinguished from sequential theory. First, the discrete composition result

$D_t^\gamma f(t)=I_t^{1-\gamma}\frac{d}{dt}f(t),$6

for the L1 discretization is an inversion result for “fractional integral $D_t^\gamma f(t)=I_t^{1-\gamma}\frac{d}{dt}f(t),$7 Caputo derivative,” not a theorem about $D_t^\gamma f(t)=I_t^{1-\gamma}\frac{d}{dt}f(t),$8 (Płociniczak, 2021). Second, high-order approximations for a single Caputo derivative, including the endpoint-corrected $D_t^\gamma f(t)=I_t^{1-\gamma}\frac{d}{dt}f(t),$9 formulas and the shifted Gegenbauer pseudospectral framework, provide building blocks that may be applied stagewise to a sequential operator, but those papers do not prove a general sequential composition law or a dedicated error theory for repeated Caputo application (Dimitrov, 2016, Elgindy, 29 Jan 2025). A plausible implication is that repeated matrix or convolution application is numerically natural, whereas its analytical order of accuracy remains a separate question.

6. Common misconceptions and current directions

The first misconception is that sequential Caputo derivatives should satisfy the same index law as fractional integrals. The supplied literature does not support that view. The integral semigroup property is explicit for $$I_t^\sigma f(t)=\frac{1}{\Gamma(\sigma)}\int_0^t (t-\xi)^{\sigma-1}f(\xi)\,d\xi.$$0, but the sequential $$I_t^\sigma f(t)=\frac{1}{\Gamma(\sigma)}\int_0^t (t-\xi)^{\sigma-1}f(\xi)\,d\xi.$$1-Caputo operator is analyzed as a composition with its own homogeneous term and Green kernel (Dien, 2021). The classical PDE treatment likewise states that $$I_t^\sigma f(t)=\frac{1}{\Gamma(\sigma)}\int_0^t (t-\xi)^{\sigma-1}f(\xi)\,d\xi.$$2 (Yusuf et al., 23 Apr 2026).

The second misconception is that all operator identities involving Caputo derivatives concern sequential differentiation. The discrete identity $$I_t^\sigma f(t)=\frac{1}{\Gamma(\sigma)}\int_0^t (t-\xi)^{\sigma-1}f(\xi)\,d\xi.$$3 and its L1 analogue are important for analysis and numerics, but they do not address Caputo-after-Caputo composition (Płociniczak, 2021).

The third misconception is that sequential Caputo derivatives can be discussed without specifying boundary or initial traces. The operator-theoretic, probabilistic, and generalized-distribution viewpoints all point in the opposite direction: domain membership, boundary correction, and singular behavior at the initial point are intrinsic to the subject [(Gorenflo et al., 2014); (Kolokoltsov, 2015); (Li et al., 2016)].

Current work therefore treats sequential Caputo derivatives as a distinct class of nonlocal operators characterized by intermediate states, additional initial data, and special-function kernels. Exact solution theory is presently most explicit for linear one-dimensional models, where bivariate Mittag-Leffler functions and Fourier reduction are available (Yusuf et al., 23 Apr 2026). Generalized boundary value theory is developed for $$I_t^\sigma f(t)=\frac{1}{\Gamma(\sigma)}\int_0^t (t-\xi)^{\sigma-1}f(\xi)\,d\xi.$$4-Caputo compositions with nonlocal conditions and weakly singular sources (Dien, 2021). Numerical practice favors sequential reformulation and single-stage discretizations applied in tandem, while a full general theory of composition, especially beyond these structured settings, remains delicate and strongly dependent on trace conditions and endpoint singularities.