Hilfer Fractional Derivative

• Hilfer fractional derivative is a two-parameter operator that interpolates between the Riemann–Liouville and Caputo derivatives through a tunable type parameter.
• It operates by applying fractional integration before and after an ordinary differentiation, yielding a controllable fading memory effect in dynamic models.
• The operator supports generalizations such as multi-order variants and kernel modifications, providing a unified framework for modeling anomalous and nonlocal dynamics.

The Hilfer fractional derivative is a two-parameter operator in fractional calculus designed to unify and interpolate between the classical Riemann–Liouville and Caputo derivatives. By introducing an additional “type” parameter, the Hilfer derivative provides a tunable framework for modeling systems with memory and hereditary properties, especially where empirical flexibility between instantaneous and lagging dynamics is required. For functions sufficiently regular on the domain of interest, the Hilfer derivative operates via fractional integration both before and after an ordinary differentiation, with the interpolation specified by the type parameter. This operator is foundational in the analysis of fractional differential equations across mathematical physics, engineering, economics, and control systems.

1. Formal Definition and Classical Limits

The left-sided Hilfer fractional derivative of order $\mu\in(0,1]$ and type $\nu\in[0,1]$ for a function $g(t)$ on $[0,\infty)$ is given by

$$D_{0+}^{\mu,\nu}g(t) = I_{0+}^{\nu(1-\mu)} \left( \frac{d}{dt} \right) I_{0+}^{(1-\nu)(1-\mu)} g(t),$$

where $I_{0+}^{\alpha}$ is the Riemann–Liouville fractional integral:

$$I_{0+}^{\alpha}g(t) = \frac{1}{\Gamma(\alpha)}\int_0^t (t-s)^{\alpha - 1} g(s)\, ds.$$

This definition reduces to the classical Riemann–Liouville derivative at $\nu=0$ and the Caputo derivative at $\nu=1$:

• For $\nu=0$: $$D_{0+}^{\mu,0}g(t) = \frac{d}{dt} I_{0+}^{1-\mu} g(t) = D_{0+}^{\mu} g(t)$$,
• For $\nu=1$: $$D_{0+}^{\mu,1}g(t) = I_{0+}^{1-\mu}\frac{d}{dt}g(t) = {}^C D_{0+}^{\mu}g(t)$$ (Raghavan et al., 2020).

2. Generalizations and Operator Variants

The Hilfer framework supports several key generalizations:

• Multi-order (bi-ordinal) Hilfer derivatives: Interpolating between derivatives of different orders and types, yielding operators such as $D_{0y}^{(\alpha,\beta)\mu}$ as in degenerate equation theory, with closed-form solutions via Kilbas–Saigo functions (Irgashev, 2023).
• Kernel generalizations: Incorporating function-dependent kernels via the $\psi$-Hilfer derivative, replacing standard powers $(t-s)^{\alpha-1}$ by $(\psi(t)-\psi(s))^{\alpha-1}$ for a strictly increasing function $\psi$ (Sousa et al., 2017, Kumbhakar et al., 2023). This recaptures a broad class including Hadamard, Katugampola, Erdélyi–Kober, and others.
• Hadamard and Katugampola variants: Using logarithmic and power kernels for more diverse singularity structures; for example, the Hilfer–Hadamard and Hilfer–Katugampola derivatives (Dhaigude et al., 2017, Salamooni et al., 2018, Oliveira et al., 2017).

3. Analytical Properties

Significant operational properties include:

• Linearity: $D_{0+}^{\mu,\nu}$ is linear.
• Laplace Transform: For appropriate $g(t)$,

$$\mathcal{L}\{D_{0+}^{\mu,\nu}g\}(s) = s^\mu \mathcal{L}\{g\}(s) - s^{\nu(\mu-1)} [I_{0+}^{(1-\nu)(1-\mu)}g](0^+)$$

facilitating exact solutions for linear fractional ODEs (Raghavan et al., 2020, Saxena et al., 2015, Plata et al., 2020).

• Memory Structure: The two integrals before and after differentiation induce a tunable two-stage fading memory effect, with the type parameter $\nu$ controlling the balance between pre- and post-differentiation history (Raghavan et al., 2020).
• Mittag–Leffler Kernel Solutions: Equations involving the Hilfer derivative admit closed-form solutions in terms of two-parameter Mittag–Leffler functions $E_{\mu,\gamma}(\cdot)$, with the Laplace pair adapted to the shifted order parameter $\gamma = \mu + \nu - \mu\nu$ (Raghavan et al., 2020). More complex variants (e.g., multinomial, Kilbas–Saigo, Prabhakar kernels) arise in higher generalizations (Salakhitdinov et al., 2017, Irgashev, 2023, Panchal et al., 2016).

4. Existence, Uniqueness, and Continuation Theory

Existence and uniqueness for Cauchy-type problems with Hilfer derivatives are established in weighted continuous spaces:

• For $n-1<\alpha<n$, $0\leq\beta\leq1$, $C_{1-\gamma}[0,T]$ spaces with $\gamma = \alpha + \beta - \alpha\beta$ capture singular initial behavior $x(t)\sim t^{\gamma-1}$ as $t\to0^+$ (Dhaigude et al., 2017).
• Volterra integral equivalence asserts that, under continuity and boundedness hypotheses on $f(t,x)$, solutions exist locally (via Schauder or Banach fixed point arguments), and extend globally provided boundedness or growth constraints are met. Uniqueness follows under Lipschitz-type conditions (Dhaigude et al., 2017, Dhaigude et al., 2017, Salamooni et al., 2018).
• For generalized operators (e.g., $\psi$-Hilfer, Hilfer–Katugampola), existence, uniqueness, and convergence properties are analogous, with explicit estimates given in terms of Mittag–Leffler functions and Picard iterations (Kucche et al., 2018, Oliveira et al., 2017, Sousa et al., 2017).
• Blow-up at the endpoint is the only obstruction to global continuation (Dhaigude et al., 2017).

5. Applications in Models and Equations

The Hilfer derivative appears in:

• Fractional economic dynamics: In the cobweb model, Hilfer's $\nu$ enables parameterized inertia in price adjustment, yielding closed-form price trajectories, rapid convergence, and smaller overshoot than either RL or Caputo cases (Raghavan et al., 2020).
• Fractional relaxation kinetics: Fractional ODEs with Hilfer derivative encompass Debye, Cole–Cole, Davidson–Cole as special cases, with closed-form, completely monotone solutions (Plata et al., 2020).
• Fractional partial differential equations: Two-term diffusion equations, hyperbolic/fractional telegraph models, and even-order Hilfer PDEs all admit exact spectral (Mittag–Leffler/kernel) solutions (Salakhitdinov et al., 2017, Irgashev, 2021, Saxena et al., 2015).
• Control and operator theory: $\psi$-Hilfer derivatives enable approximate controllability in Banach/Hilbert spaces, with solution representations via operator families and mild solution formulas (Kumbhakar et al., 2023, Jaiswal et al., 2019).
• Numerics and boundary value problems: Bernstein-spline techniques, tailored to the singular initial behavior, achieve convergence rates $O(h^\alpha + q^{-\alpha/2})$ for Hilfer-IVPs and fractional-periodic boundary conditions, even in nonlinear settings (Goedegebure et al., 28 Mar 2025, Goedegebure et al., 20 Jan 2026).

6. Generalizations and Unification via Kernel Operators

The Hilfer construction serves as the foundation for broader operator families:

• $\psi$-Hilfer derivative: For $\psi\in C^n$, $$^{H}D_{a+}^{\alpha,\beta;\psi}f(x) = I_{a+}^{\beta(n-\alpha);\psi}(1/\psi'(x) d/dx)^n I_{a+}^{(1-\beta)(n-\alpha);\psi}f(x)$$. By choosing $\psi(x)=x$, $\psi(x)=\ln x$, or more exotic $\psi$, one recovers classical, Hadamard, Katugampola, Erdélyi–Kober, Prabhakar-type, and many other operators (Sousa et al., 2017).
• Hilfer–Hadamard and Hilfer–Katugampola: These interpolate between Hadamard/Caputo–Hadamard and Katugampola/Caputo–Katugampola, respectively, and subsume Liouville, Weyl, Feller, Prabhakar, and related derivatives (Dhaigude et al., 2017, Salamooni et al., 2018, Oliveira et al., 2017).
• Leibniz-type product rules: Extended to the general $\psi$-Hilfer and other kernels, yielding product differentiation formulas crucial for variational and stability analyses (Sousa et al., 2018).

7. Significance, Advantages, and Modeling Implications

By continuous variation of the type parameter ($\nu$, $\beta$, or more generally $\Psi$-parameterizations), the Hilfer derivative interpolates between classical fractional derivatives and adapts the physical, economic, or engineering model to observed data with tunable memory effects. Key modeling consequences include:

• Enhanced flexibility in empirical fitting (e.g., fractional inertia in price or relaxation kinetics).
• Analytic tractability via Laplace/Mittag–Leffler representations.
• Stable and convergent numerical approximations handling singular initial data.
• Unified treatment of fractional operators arising in heterogeneous, anomalous, or nonlocal systems (Raghavan et al., 2020, Goedegebure et al., 28 Mar 2025, Goedegebure et al., 20 Jan 2026, Plata et al., 2020).

For rigorous theory, implementation, and applications, the Hilfer derivative and its variants are indispensable in contemporary fractional calculus, offering deep connections between analysis, operator theory, applied modeling, and computational methods.