General Fractional Derivatives (GFDs)

• General Fractional Derivatives (GFDs) are extensions of classical differentiation that modify kernels, clocks, or function spaces to incorporate nonlocal memory effects.
• They encompass various formulations including Sonine-kernel, parametric, and local limit derivatives, each offering unique analytic and numerical advantages.
• Their broad applicability spans fractional differential equations, anomalous diffusion, and analytic function spaces, with open challenges in stability and equivalence.

In the literature surveyed here, “general fractional derivative” (GFD) does not denote a single operator. It is an umbrella term for several distinct generalizations of differentiation that preserve, extend, or reinterpret aspects of fractional calculus by modifying the kernel, the composition structure, the underlying variable, or the ambient function space. The most developed families include Sonine-kernel convolution operators of Riemann–Liouville and Caputo type, parametric operators with respect to another function $g$, local limit-defined derivatives controlled by a kernel or increment map, Katugampola’s $\rho$-dependent unification of Riemann–Liouville and Hadamard calculus, and Bergman-kernel-induced operators on analytic function spaces (Luchko, 2022, Tarasov, 2 Sep 2025, Akkurt et al., 2017, Katugampola, 2011, Perälä, 2018).

1. Terminological scope and basic classification

A central structural distinction is between nonlocal GFDs, defined by Volterra-type convolution or other integral kernels, and local GFDs, defined by a weighted first derivative or a limit of generalized increments. In the first class, memory enters through kernels such as power laws, tempered kernels, Mittag–Leffler kernels, Bessel-type kernels, or Sonine pairs; in the second, the operator often takes the form of a pointwise multiplier applied to the ordinary derivative. A further distinction concerns the underlying geometry: some GFDs act “with respect to another function” $g$, some interpolate between additive and multiplicative calculus, and some are defined by reproducing kernels rather than time-memory kernels (Tarasov, 2 Sep 2025, Mingarelli, 2018, Katugampola, 2011, Perälä, 2018).

On finite intervals, the associated fractional integrals are much less ambiguous than the derivatives. Under the conditions stated by Cartwright and McMullen, there exists precisely one unique family of the fractional integrals, namely the Riemann–Liouville fractional integrals, whereas the left-inverse problem for those integrals admits infinitely many derivative families (Luchko, 2020). This asymmetry is one reason the term GFD covers a broad and genuinely plural literature.

Framework	Defining mechanism	Representative sources
Sonine-kernel GFDs	Convolution kernels $K,k$ with $K*k=1$ or $K*k=h_n$	(Luchko, 2021, Luchko, 2022, Luchko, 2021)
Parametric GFDs	Operators with respect to another function $g$ or via scale/weight functions	(Tarasov, 2 Sep 2025, Kumar et al., 2022)
Local GFDs	Limit-defined weighted derivatives such as $m_\alpha(t)f'(t)$	(Akkurt et al., 2017, Mingarelli, 2018, Toghani et al., 2021, Abu-Shady et al., 2021)
Unified RL–Hadamard GFDs	$\rho$-dependent integral and derivative pair	(Katugampola, 2011)
Analytic-function-space GFDs	Coefficient multipliers induced by weighted Bergman kernels	(Perälä, 2018)

This plurality also explains recurring disagreements over semigroup laws, admissible initial data, and the meaning of “fractional order.” In some frameworks, order is encoded by a singular kernel exponent; in others, by an auxiliary type parameter, a nonlinear clock $g$, or a reproducing-kernel deformation.

2. Sonine-kernel and convolutional general fractional calculus

The most systematic nonlocal theory is based on Laplace convolution

$\rho$0

together with Sonine-type kernel relations. For kernels $\rho$1 satisfying $\rho$2, the general fractional integral is

$\rho$3

the Riemann–Liouville-type GFD is

$\rho$4

and the Caputo-type GFD is

$\rho$5

which becomes $\rho$6 on $\rho$7 (Luchko, 2022). In the arbitrary-order extension, the generalized Sonine condition is

$\rho$8

and the RL-type operator becomes

$\rho$9

with a Caputo-type regularization obtained by subtracting the first $g$0 Taylor terms of $g$1 at the origin (Luchko, 2021).

Within this framework, the classical Abel kernels $g$2 and $g$3 recover the usual Riemann–Liouville and Caputo operators. The same formalism also accommodates Bessel-type Sonine pairs, Mittag–Leffler / Hanyga-type pairs, Zacher-type modifications, tempered kernels, and other non-power-law memories (Luchko, 2022). The spaces $g$4, $g$5, and $g$6 are used to control admissible singularities at $g$7 and to formulate the operator identities rigorously (Luchko, 2021, Luchko, 2021).

The decisive structural fact is that the generalized fractional integral and derivative satisfy analogues of the two fundamental theorems of fractional calculus. In the order-below-one Sonine setting one has

$g$8

on the appropriate spaces, and

$g$9

under the stated domain assumptions (Luchko, 2022). For arbitrary order $K,k$0, the Caputo-type second fundamental theorem becomes

$K,k$1

which makes the role of classical initial data explicit (Luchko, 2021).

A major refinement is the “1st level” construction

$K,k$2

where the kernel triple $K,k$3 satisfies $K,k$4. This operator contains RL-type GFDs as the case $K,k$5, Caputo-type GFDs as the case $K,k$6, and Hilfer’s derivative as the power-law example $K,k$7 (Luchko, 2022). Its second fundamental theorem introduces the projector term

$K,k$8

so the “initial value” is neither $K,k$9 nor $K*k=1$0 alone, but a kernel-dependent quantity interpolating between the two classical paradigms (Alkandari et al., 2024).

In the RL sense on finite intervals, the arbitrary-order GFDs possess explicit null spaces and projector operators. If $K*k=1$1, then the kernel of the RL-type GFD is spanned by derivatives of $K*k=1$2,

$K*k=1$3

and the second fundamental theorem yields a closed-form projector

$K*k=1$4

which defines the natural initial conditions for RL-type problems (Luchko, 2022).

The Mikusiński-type operational calculus built on the convolution quotient field then converts many linear GFD equations into algebraic equations in the symbol $K*k=1$5, with resolvent kernels

$K*k=1$6

For power-law kernels this reproduces the familiar Mittag–Leffler resolvents, while for general Sonine kernels it yields exact convolution-series solutions for single-term and multi-term Cauchy problems (Alkandari et al., 2024, Luchko, 2021).

3. Parametric operators and differentiation with respect to another function

A second major strand defines GFDs relative to a strictly increasing $K*k=1$7 function $K*k=1$8 with $K*k=1$9. In the canonical RL-type form,

$K*k=h_n$0

while the Caputo-type form is

$K*k=h_n$1

with left- and right-sided versions defined through kernels depending on $K*k=h_n$2 and $K*k=h_n$3, respectively (Tarasov, 2 Sep 2025). These operators are not ad hoc deformations: via the substitution operator $K*k=h_n$4, they are conjugates of ordinary general fractional operators on the transformed interval $K*k=h_n$5. That is the basis for their semigroup laws, inverse relations, and integration-by-parts formulas.

Special choices of $K*k=h_n$6 recover standard named calculi. The choice $K*k=h_n$7 gives the classical RL and Caputo operators; $K*k=h_n$8 yields the Hadamard case with $K*k=h_n$9; and $g$0 produces Erdélyi–Kober-type forms after change of variables (Tarasov, 2 Sep 2025). The same paper gives explicit formulas such as

$g$1

and

$g$2

showing that the standard algebra of power kernels survives after replacing the linear clock by $g$3.

A related but distinct parametric formulation uses a scale function $g$4 and a weight function $g$5 in a Caputo-type GFD for diffusion equations. In that setting, $g$6 is strictly increasing, $g$7, and the operator acts on the weighted field $g$8 through the deformed scale variable $g$9. Choosing $m_\alpha(t)f'(t)$0 and $m_\alpha(t)f'(t)$1 reduces the model to the standard Caputo derivative; appropriate other choices encompass Riemann–Liouville, Riesz, and Hadamard derivatives in particular cases (Kumar et al., 2022). For the resulting generalized fractional diffusion equations, a higher-order finite difference scheme was derived with second-order spatial accuracy and temporal accuracy $m_\alpha(t)f'(t)$2, and the paper emphasizes the modeling role of $m_\alpha(t)f'(t)$3 as a time-rescaling device and $m_\alpha(t)f'(t)$4 as a history-reweighting factor (Kumar et al., 2022).

Both versions are motivated by nonuniform memory. In the $m_\alpha(t)f'(t)$5-calculus, the past is sampled in the variable $m_\alpha(t)f'(t)$6, so $m_\alpha(t)f'(t)$7 acts as a nonlinear operational time. In the $m_\alpha(t)f'(t)$8 formulation, the memory kernel is reshaped simultaneously by scale and amplitude. This suggests a general principle: many “new” GFDs are best understood as classical fractional operators transported by a nonlinear clock or a weighted similarity transform.

4. Local and limit-defined general fractional derivatives

Another large body of work uses the term GFD for local operators defined by generalized increments. One formulation introduces a continuous nonnegative kernel $m_\alpha(t)f'(t)$9 with $\rho$0 and $\rho$1 for $\rho$2, and defines the derivative by a limit involving an exponential perturbation of the argument. For differentiable $\rho$3, the fundamental identity is

$\rho$4

which immediately yields linearity, product rule, quotient rule, chain rule, Rolle’s theorem, the mean value theorem, and inverse relations with a matching generalized fractional integral (Akkurt et al., 2017). This framework is explicitly local: it does not introduce a memory kernel, and the paper does not provide Laplace/Fourier or nonlocal-kernel representations.

A more general local scheme is based on an increment map $\rho$5 and the limit

$\rho$6

Under suitable nondegeneracy conditions, $\rho$7, so the operator is again a weighted ordinary derivative (Mingarelli, 2018). Its principal interpretation is geometric: after the time reparametrization

$\rho$8

one gets $\rho$9. This is why, in the mechanics examples treated there, generalized oscillators, central-force orbits, and even the Newtonian $g$0-body problem retain their classical orbit geometry, with the generalized derivative affecting only the time parametrization (Mingarelli, 2018).

A closely related algebraic formulation writes

$g$1

with $g$2 chosen so that various conformable-type derivatives become special cases. For $g$3, the pair $g$4 forms a differential ring, so the operator obeys the ordinary Leibniz and quotient rules (Toghani et al., 2021). The price of this simplicity is that the operator is local unless $g$5 is allowed to depend on the full history of $g$6, in which case linearity and locality are lost.

A particularly explicit variant is

$g$7

introduced for $g$8 and $g$9 (Abu-Shady et al., 2021). On monomials, with the parameter matched termwise, this reproduces the standard RL/Caputo coefficient

$\rho$00

and for Taylor-expandable functions the paper proves

$\rho$01

The same paper uses this locality to solve Riccati-type fractional differential equations in elementary closed form and reports lower errors than the conformable derivative in the benchmark problems considered there (Abu-Shady et al., 2021).

These local GFDs therefore occupy a different conceptual niche from RL-, Caputo-, or Sonine-kernel operators. They preserve ordinary differential algebra almost exactly, but they do so by replacing memory with a weighted first derivative or a clock change. A recurring misconception is to treat them as interchangeable with nonlocal fractional derivatives; the literature itself emphasizes that such an identification fails whenever genuine history dependence is essential (Mingarelli, 2018).

5. Specialized unifications, reproducing-kernel models, and equivalence issues

Several influential constructions generalize fractional differentiation by changing the underlying analytic model rather than merely the kernel. Katugampola’s unified derivative begins from the generalized integral

$\rho$02

and defines

$\rho$03

As $\rho$04, this reduces to the Riemann–Liouville operators; as $\rho$05, it reduces to the Hadamard operators (Katugampola, 2011). The same paper proves semigroup, composition, and inversion properties and gives the power-function formula

$\rho$06

which makes the RL and Hadamard limits transparent (Katugampola, 2011).

A different unifying proposal starts from the translation operator $\rho$07 and defines

$\rho$08

with a constitutive function $\rho$09 evaluated at the shift operator. Choosing $\rho$10 yields the generalized binomial difference

$\rho$11

from which the Grünwald–Letnikov and Marchaud derivatives are obtained, and under suitable hypotheses one recovers Riemann–Liouville, Weyl, Riesz, and Caputo variants as particular cases (Michelitsch et al., 2011). This framework is notable because it makes nonlocality a direct consequence of the singular behavior of $\rho$12 at $\rho$13.

The relation between RL and Grünwald–Letnikov differentiation is itself subtler than a simple equality statement suggests. For the “generalised Cauchy derivative” — identified in the paper with the Riemann–Liouville derivative — the GL infinite sum may diverge even when the RL derivative exists. The proposed resolution is a principal-value interpretation in which the GL partial sum is truncated at $\rho$14 and the mesh size is coupled to $\rho$15 through

$\rho$16

For $\rho$17, the characteristic equation is

$\rho$18

with $\rho$19 always a solution, and for Taylor-expandable functions the universal choice

$\rho$20

recovers the RL value (Sudhir, 2018). This result is both an equivalence theorem and a warning: path dependence in the $\rho$21-limit matters.

Outside time-domain calculus, “general fractional derivative” also names an operator on analytic function spaces induced by weighted Bergman kernels. For radial doubling weights $\rho$22, the operator $\rho$23 is defined by the kernel transformation rule $\rho$24 and acts on Taylor coefficients by

$\rho$25

It satisfies

$\rho$26

so it is an isomorphism on the space $\rho$27 of analytic functions on the disk (Perälä, 2018). In this context, “fractional derivative” means a coefficient multiplier asymptotic to $\rho$28 on monomials when $\rho$29, and the main applications concern surjectivity of weighted Bergman projections onto Bloch and Besov spaces (Perälä, 2018).

These specialized constructions show that GFDs are not merely a proliferation of kernels. They also arise from Mellin-type geometry, translation-operator renormalization, reproducing-kernel theory, and subtle limit procedures connecting formally equivalent definitions.

6. Applications, numerical schemes, and open problems

The application landscape reflects the same diversity. In the Sonine-kernel setting, the operational calculus leads to closed-form solutions of linear fractional differential equations, including single-term and multi-term initial value problems, with generalized resolvent kernels extending exponentials and Mittag–Leffler functions (Alkandari et al., 2024). The parametric calculus with respect to $\rho$30 is explicitly motivated by memory processes, anomalous relaxation and transport, viscoelastic constitutive laws, and even economic interpretations in which $\rho$31 is chosen as an economic factor or $\rho$32 (Tarasov, 2 Sep 2025). The broader program of “general fractional dynamics” pushes the same kernel logic into exact discrete-time nonlocal maps for periodically kicked systems, where the map weights are determined directly by Sonine/Luchko kernels rather than introduced phenomenologically (Tarasov, 22 Sep 2025).

For PDEs, the scale-weight formulation of generalized fractional diffusion yields a higher-order finite difference scheme for generalized fractional diffusion equations. The paper reports second-order spatial convergence and temporal order $\rho$33, and uses nonlinear scale functions and nonconstant weights to model heterogeneous time scales and reweighted memory (Kumar et al., 2022). This suggests that numerical analysis for GFDs is best understood as the joint discretization of a kernel and a geometry: the same formal order $\rho$34 can behave very differently depending on whether the variability is placed in $\rho$35, $\rho$36, $\rho$37, or $\rho$38.

In complex analysis, Bergman-kernel GFDs supply constructive preimages for weighted Bergman projections onto Bloch and Besov spaces, even when the target space is not contained in the domain space. The formulas

$\rho$39

make the surjectivity statements explicit (Perälä, 2018). In mechanics, the local increment-map derivatives act as time changes, so generalized harmonic oscillators, Hooke-type central-force trajectories, and Newtonian $\rho$40-body dynamics preserve their classical orbit geometry modulo reparametrization (Mingarelli, 2018).

The open problems are correspondingly framework-specific. In the RL–GL principal-value theory, the paper explicitly highlights the problem of determining characteristic polynomials and admissible roots for general polynomials beyond the universal solution $\rho$41 (Sudhir, 2018). In the $\rho$42-parametric calculus, broader kernel classes, nonmonotone $\rho$43, and numerical stability/error bounds for complex kernels remain open (Tarasov, 2 Sep 2025). In Bergman-space GFDs, non-radial weights and sharper norm formulas are left open (Perälä, 2018). In the local kernel-based model of (Akkurt et al., 2017), semigroup laws, transform-domain formulas, and recovery of classical nonlocal operators beyond conformable-style cases are explicitly not provided.

Taken together, these developments support a precise but nonunitary conclusion. General fractional derivatives form a family of operator theories rather than a single doctrine. What unifies them is not one definition, but a recurring program: to preserve some core feature of fractional calculus — inverse relations, interpolation between known operators, generalized memory kernels, or analytically tractable functional identities — while relaxing the classical assumptions on kernel, clock, domain, or algebraic structure.