Discrete Fractional Variants: Theory & Applications

• Discrete fractional variants are discrete analogues of fractional calculus that generalize difference operators to noninteger orders, incorporating memory and nonlocal effects.
• They employ operator frameworks like Riemann–Liouville, Caputo, and Grünwald–Letnikov to facilitate numerical methods and variational formulations in discrete systems.
• Applications include fractional discrete Laplacians, optimal control, chaotic maps, and stochastic processes, offering rich insights into spectral properties and long-range interactions.

Discrete fractional variants encompass a spectrum of discrete analogues of fractional calculus—extensions of integer-order derivatives and integrals to noninteger orders—systematically incorporating memory and nonlocality into discrete systems. These variants include discrete fractional sums and differences (of Riemann–Liouville, Caputo, binomial, and Grünwald–Letnikov types), generalized operator frameworks, and fractionalizations embedded into difference equations, equations of motion, variational principles, and stochastic processes. In addition, fractional discrete Laplacians are used to induce long-range spatial coupling in lattice models, interpolating from local to global interactions as the fractional order varies. This article synthesizes the foundational frameworks, canonical operator families, numerical methods, representative applications, and spectral structures of discrete fractional variants as developed in the modern literature.

1. Operator Frameworks for Discrete Fractional Calculus

Discrete fractional calculus generalizes the difference operator and discrete summation to noninteger orders, leading to discrete analogues of classical Riemann–Liouville and Caputo operators. Canonical constructions fall into two broad classes:

a) Riemann–Liouville and Caputo Discrete Fractional Operators:

• The left discrete Riemann–Liouville fractional sum is

$${}_{a}\Delta_t^{-\nu}f(t) =\frac{1}{\Gamma(\nu)} \sum_{s=a}^{t-\nu} (t-\sigma(s))^{(\nu-1)}\,f(s),$$

with $t$ in a suitable lattice. The fractional (difference) derivative is defined as $\Delta$ of the sum (Bastos et al., 2010).

• Caputo-type fractional differences move the discrete difference operator inside the fractional sum, altering admissible boundary conditions (Abdeljawad, 2017).

b) Grünwald–Letnikov and Binomial Approaches:

• Grünwald–Letnikov (GL) discrete derivatives use convolution sums with binomial coefficients:

$$D_h^\alpha x(t_i) = h^{-\alpha} \sum_{k=0}^i \omega_k^{(\alpha)} x_{i-k},$$

where $\omega_k^{(\alpha)} = (-1)^k \binom{\alpha}{k}$. Binomial/GL and Riemann definitions are equivalent via explicit duality identities (Abdeljawad et al., 2012).

c) Generalized Convolution-Based Operators:

• Ferreira's convolutional framework (Ferreira, 2021) defines a generalized fractional sum via a sequence $p$:

$$\mathcal{S}_{\{(p);\,a\}} f(t) = \sum_{\tau=a}^{t-1} p(t-\tau-1+a)\,f(\tau),$$

and generalized differences via sequence $q$, such that $(p*q)_a(t) = 1$. All known fractional delta/nabla operators are encompassed as special cases by suitable $p,q$.

These operator frameworks extend naturally to nabla (backward) differences and time scales, including $t$0 and multi-dimensional lattices (Bastos, 2012).

2. Discrete Fractional Variational Calculus

Discrete fractional variants underpin direct numerical methods in the calculus of variations, fractional optimal control, and fractional Sturm–Liouville theory:

• Euler–Lagrange equations for functionals with discrete fractional differences generalize the classical necessary conditions. For example, for a left fractional difference of order $t$1 and right fractional difference of order $t$2, the Euler–Lagrange equation reads:

$t$3

(Bastos et al., 2010, Abdeljawad, 2017, Bastos et al., 2010)

• Legendre-type second-order conditions are established for discrete fractional systems to rule out spurious extremals (Bastos et al., 2010, Bastos et al., 2010, Bastos, 2012).
• Direct discrete methods use GL approximations to convert fractional variational problems to finite-dimensional optimization, facilitating practical computation on grids. The resulting discrete Euler–Lagrange systems are nonlinear algebraic equations for nodal values, with first-order convergence in mesh size (Pooseh et al., 2012, Pooseh et al., 2013).
• Fractional Sturm–Liouville problems can be discretized via fractional difference operators:

$t$4

leading to generalized eigenvalue problems with self-adjoint and spectral properties analogous to the continuous case (Almeida et al., 2016).

• Discrete fractional optimal control is codified via fractional Pontryagin systems, with well-posed discrete variational integrators and Noether theorems providing exact conservation laws in the presence of discrete symmetries (Bourdin, 2012).

3. Fractional Discrete Maps and Dynamical Systems

Discrete fractional variants yield maps and dynamical systems with intrinsic long-range memory:

• Fractional Caputo and Riemann–Liouville difference equations generate maps where each subsequent value encodes a power-law weighted sum over all prior iterates (falling factorial–law memory) (Edelman, 2014):

$t$5

• Fractional kicked systems (universal, standard, dissipative maps) generalize classical maps by embedding fractional derivatives in either the flow or the kick, leading to discrete convolution kernels whose weights decay as power laws. For instance, the fractional standard (Chirikov) map:

$t$6

exhibits non-Markovian phase-space dynamics and fractional chaotic attractors (Tarasov et al., 2011, Tarasov, 2011).

• Memory effects: The algebraic decay of the memory kernel fundamentally alters bifurcation cascades, stability, and attractor properties compared to their integer-order analogues (Edelman, 2014).

4. Fractional Discrete Laplacians and Lattice Models

Fractional discrete Laplacians introduce tunable long-range coupling into lattice systems, leading to discretized analogues of the Riesz fractional operator:

• In 1D, for $t$7,

$t$8

with $t$9 explicit in terms of Gamma functions (Molina, 2019).

• In 2D square lattices, the discrete fractional Laplacian is constructed via spectral/semigroup representations, leading to kernels $\Delta$0 with "faster than exponential" decay and an effective site energy (Mejía-Cortés et al., 2021).
• Fractional discrete nonlinear Schrödinger (FDNLS) equations:

$\Delta$1

interpolate between local (nearest-neighbor, $\Delta$2) and all-to-all ($\Delta$3) coupling, with resulting consequences for spectrum flattening, linear transport, soliton width, and self-trapping transitions (Molina, 2019).

• Discrete vortex solitons in 2D FDNLS models persist for all $\Delta$4; smaller $\Delta$5 lowers the power threshold for stability and promotes long-range mediated stabilization of helical modes (Mejía-Cortés et al., 2021).

5. Applications to Stochastic Processes and Fractional Poisson Models

Generalizations of compound/mixed Poisson processes in discrete time admit fractionalization via random time (or space) change with stable subordinators:

• Time-fractional Poisson processes $\Delta$6 for $\Delta$7 have pmf given by:

$\Delta$8

with $\Delta$9 the two-parameter Mittag–Leffler function (Beghin et al., 2013).

• For compound processes $$D_h^\alpha x(t_i) = h^{-\alpha} \sum_{k=0}^i \omega_k^{(\alpha)} x_{i-k},$$0, the fractional Kolmogorov equations replace time derivatives with Caputo derivatives of order $$D_h^\alpha x(t_i) = h^{-\alpha} \sum_{k=0}^i \omega_k^{(\alpha)} x_{i-k},$$1.
• Overdispersion is enhanced in the fractional case compared to the classical Poisson process, and fractional negative binomial and Pólya–Aeppli processes are constructed analogously (Beghin et al., 2013).

6. Discrete Fractional Transforms and Spectral Methods

The discrete fractional Fourier transform (DFrFT) generalizes the discrete Fourier transform (DFT) to arbitrary (fractional) orders, with applications in signal processing and quantum optics:

• DFrFTs may be realized by propagations in $$D_h^\alpha x(t_i) = h^{-\alpha} \sum_{k=0}^i \omega_k^{(\alpha)} x_{i-k},$$2-lattices (engineered waveguide arrays), with explicit kernels given in terms of Jacobi polynomials, and fractional orders corresponding to longitudinal position on the chip (Weimann et al., 2015).
• The shift theorem under DFrFT mirrors the DFT case, introducing a phase ramp proportional to both shift and fractional order.
• Quantum generalizations act naturally on multi-photon and entangled states, yielding analytic expressions for coincidence matrices (Weimann et al., 2015).

7. Advanced Topics and Generalizations

• Generalized discrete fractional operators: The convolutional framework (Ferreira, 2021) unifies classical, binomial, and discrete fractional sums and differences by constructing all operators from a pair of kernels $$D_h^\alpha x(t_i) = h^{-\alpha} \sum_{k=0}^i \omega_k^{(\alpha)} x_{i-k},$$3 satisfying a convolution identity.
• Fractional Hadamard derivatives: Discrete approximations for the Hadamard fractional derivative yield high-order, nonlocal finite difference schemes with explicit convergence rates and error bounds, enabling efficient variational and ODE solvers (Almeida et al., 2016, Almeida et al., 2014).
• Time scales fractional calculus: Discrete fractional calculus naturally extends to arbitrary time scales, with Laplace-transform-based definitions and convolutional semigroup laws, encompassing both $$D_h^\alpha x(t_i) = h^{-\alpha} \sum_{k=0}^i \omega_k^{(\alpha)} x_{i-k},$$4 and $$D_h^\alpha x(t_i) = h^{-\alpha} \sum_{k=0}^i \omega_k^{(\alpha)} x_{i-k},$$5 in a unified formalism (Bastos, 2012).

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These discrete fractional variants form a coherent theoretical and computational foundation for modeling memory, nonlocality, and anomalous phenomena across discrete, lattice, and time-scale systems, with far-reaching applications in numerical analysis, physics, engineering, and stochastic modeling. Foundational frameworks, such as those by Ferreira (Ferreira, 2021), Bastos–Ferreira–Torres (Bastos et al., 2010, Bastos et al., 2010), Pooseh–Almeida–Torres (Pooseh et al., 2012, Pooseh et al., 2013), and explicit fractional Laplacians (Mejía-Cortés et al., 2021, Molina, 2019), provide the essential mathematical infrastructure underpinning current research in this domain.