Building on the recently published framework of advanced tensor theories—where tensors are rigorously defined via standard and fractional derivatives—this work extends the formalism to foundational mathematics, fractional calculus, and fractional geometry. We introduce these advanced tensors as generalized objects that unify differential structures across integer and non-integer orders, enabling seamless interpolation between classical and fractional regimes.

Central to the presentation is the construction of transformation laws for mathematical objects under advanced tensor actions, revealing that possibly novel algebraic and geometric invariants are preserved across fractional dimensions. These transformations are shown to induce natural fractional differential geometries on manifolds, yielding fractional manifold–metric pairs with curvature expressions involving Caputo–Fabrizio or Riemann–Liouville operators.

Real-world applications are explored in depth: modeling anomalous diffusion in heterogeneous media, designing fractional physical and engineering systems with memory- and environment-dependent dynamics, and developing scale-invariant image processing algorithms using fractional tensor convolutions. We further demonstrate how advanced tensors facilitate multi-scale physics simulations—bridging quantum microstates to macroscopic continuum behavior—through dimensionally hybrid tensor fields.

By establishing rigorous links between abstract fractional structures and actionable mathematical, physical, and engineering paradigms, this study positions advanced foundational tensors as a unifying language for next-generation mathematical modeling of sophisticated and complex systems, such as comology, gravity, and high-energy physics.

The study of complex dynamical systems exhibiting memory effects, anomalous diffusion, or multi-scale relaxation has grown significantly across physics, electrochemistry, porous media, and biology. Experimental evidence shows that many real-world systems do not follow classical exponential laws or standard integer-order diffusion models. This has motivated the development of alternative mathematical frameworks capable of capturing long-range temporal correlations and broad relaxation spectra.

Fractional-order models have become popular because they naturally produce power-law memory kernels and can reproduce several experimentally observed behaviors. However, their use now appears increasingly problematic. Major limitations include: lack of a clear physical interpretation of fractional parameters, structural non-uniqueness, numerical difficulties, multiplicity of definitions of fractional derivatives, physical inconsistency of initial conditions, and the risk of misinterpreting fractional behavior.

These issues call for caution and for systematic criteria to determine whether a system truly requires fractional-calculus-based modeling. In many cases found in the literature, such justification is absent.

In this context, two alternative approaches, the Distribution of Relaxation Times (DRT) and diffusive representations, provide physically interpretable and mathematically transparent modeling tools. Both rely on expressing the system response as a continuous superposition of exponentially decaying modes, and they can be seen as equivalent formulations of the same underlying spectral framework.

The article therefore aims to (1) clarify the theoretical links between DRT, diffusive representation, and infinite-state models; (2) show on several examples from the literature that DRT can rigorously distinguish genuinely fractional behaviors from behaviors that are not fractional (to avoid using fractional models unjustifiably); (3) demonstrate how DRT enables direct time-domain modeling from experimental data produced by a gas sensor with comparable accuracy and as few parameters as a fractional model; and (4) illustrate how DRT-derived kernels yield accurate and physically meaningful models. Overall, the goal is to promote DRT and diffusive representations as robust alternatives to fractional-order models.

Intra-articular drug injection is a core therapeutic approach for knee osteoarthritis (KOA), and its efficacy is closely related to the spatial distribution of the injected fluid. Supported by fractional-order derivative theory, this study first constructs an anatomically accurate model of the human knee joint cavity as a foundation for flow simulation. Given that the injected drug exhibits time-dependent viscoelastic relaxation with pronounced memory effects, the fractional Maxwell constitutive equation is adopted to more accurately describe its mechanical behavior. This constitutive model is then embedded in the fluid-dynamic governing equations, which are discretized and solved using the finite volume method combined with the L1 algorithm. Through numerical simulations, we systematically investigate how key parameters—such as the characteristic relaxation time and the fractional derivative order—affect the intra-articular distribution of the injected drug. The results indicate that increasing the fractional derivative order enhances the effective distribution volume of the drug within the joint cavity and increases its retention in peripheral regions, while reducing distribution uniformity in the central region. By introducing fractional-order derivatives, this study enables a more precise representation of drug flow behavior during injection. The results elucidate the regulatory roles of key parameters in shaping drug distribution, thereby offering valuable scientific guidance for the development of personalized KOA injection-based treatment strategies.

This paper presents an advanced analysis of multivariable control loops applied to industrial processes using variable-order fractional calculus. The study focuses on a Multi-Input Multi-Output (MIMO) system represented by the classical Wood & Berry model of a binary distillation column. This benchmark model is characterized by cross-coupling and time delays, relating the distillate (X_D) and bottoms (X_B) compositions to the manipulated variables: reflux flow rate and reboiler heat duty.

To address the challenges of these coupled dynamics, the proposed control strategy employs Fractional Order PI controllers (Gc₁ and Gc₂) based on the Scarpi variable-order fractional integral. The fractional order of the integral part of the controller, α(t), is modeled as a time-varying exponential function rather than a fixed value, allowing the controller to adapt its aggressiveness over time. To obtain the time-domain response, the closed-loop equations formulated in the Laplace domain were numerically inverted. While various inversion methods are reported in the literature, this work used the method proposed by de Hoog et al., implemented via the invertlaplace function from the Python MpMath package (Version 1.2.1).

The controller parameters were estimated by minimizing the Integral of Squared Error (ISE), resulting in a total objective function value of 4.04. The transient response analysis indicates a settling time of approximately 60 time units. The variable-order strategy successfully decoupled the system, suppressing the interaction peak on the bottoms composition ($X_B$) and returning it to the set-point, demonstrating robust disturbance rejection capabilities.

Introduction: Recent research has demonstrated that nanofluids exhibit a nonlinear relationship between their shear stress and shear rates due to the differing particle characteristics. Some complex classical mechanical models have been employed to describe the rheological properties of nanofluids. Given the heterogeneity of nanofluids, anomalous diffusion, Brownian motion-induced nano-convection, nanoparticle clustering, and thermophoresis often occur, leading to a large disparity between simulation results from classical mathematical models and experimental measurements. The traditional rheological constitutive equations may not be suitable for describing these anomalous phenomena; however, they can be effectively addressed using fractional calculus.

Methods: Justification for the use of fractional calculus in nanofluid modelling is discussed from the viscoelastic mechanical viewpoint. The well-established models, including Newtonian, Maxwell, Kelvin-Voigt, and anti-Zener models, are summarised and generalised to the fractional setting. Several classes of convolution kernel functions are derived, which are suitable for describing the rheological behaviour of viscoelastic nanofluids. The effects of using different fractional operators in the constitutive equation of nanofluids are also investigated.

Results: It is found that the Maxwell constitutive equation with the Caputo-Fabrizio or Atangana-Baleanu-Caputo derivative is a special case of the generalised constitutive model with the fractional Riemann-Liouville derivative. Validation of the fractional Maxwell constitutive model against experimental data of nanofluids demonstrates that the proposed fractional constitutive formulation can effectively capture the rheology of non-Newtonian nanofluids exhibiting shear-thinning type curves. Based on a novel development of the nonlocal constitutive relationship, a unifying generalised nanofluid model incorporating rheological and heat relaxation is proposed.

Conclusions: This study provides useful insights into treating nanofluids as a class of viscoelastic fluids using fractional calculus and highlights the potential of employing nonlocal mathematical models to simulate nanofluid flow in real applications.

Modeling fractional behaviors is fundamental for characterizing complex phenomena governed by power-law dynamics, long-range temporal and spatial correlations, fractal structures, and memory effects, which are commonly observed in diverse scientific and engineering applications. In this work, we provide an analytical derivation of a fractional formulation for steady-state heat-conduction modeling. The boundary-value problem is defined on an annular domain, with boundary conditions prescribed as functions of the angular coordinate. The proposed approach integrates two analytical techniques, namely separation of variables and fractional power-series expansion, within a framework based on the Caputo fractional derivative. Beyond heat conduction, the methodology presented here is also directly applicable to a broader class of boundary-value problems governed by elliptic fractional differential equations, supporting advancements in modeling heterogeneous materials, porous media, biological tissues, and systems with multiscale or memory-driven behavior. A distinguishing aspect of this study is its emphasis on analytical transparency and didactic clarity, with each mathematical step introduced and developed systematically. The resulting closed-form solution provides a useful reference for future investigations into fractional differential equations, offering both methodological guidance and conceptual insight. The results include the explicit solution in polar coordinates and its interpretation as a function of an arbitrarily selected fractional order, thereby illustrating how fractional operators modulate steady-state heat conduction under the specified boundary conditions.

Triangular Sierpinski structures are fractal geometries that have recently been studied for microwave frequency applications and implemented in microstrip and coplanar waveguide configurations. Antennas and planar components can be proposed using the Sierpinski geometry. The main advantage of the above solution is the definition of a building block with increased internal complexity to modulate the operating frequency within a frequency range defined by the triangle's edge length, set as a fixed starting parameter. Increasing the complexity of the internal geometry by the number of sub-triangles inside the original figure helps define a shifted resonance frequency, thereby enabling frequency tuning by geometry and, through coupled structures, enhancing the electrical performance of the single triangle via edge coupling. A wider bandwidth is achieved by coupling Sierpinski triangles with increased internal complexity. The single building block must be appropriately excited to obtain the primary resonance frequency and the higher-order modes expected for the entire, non-subdivided triangle. Then, a spectral analysis is mandatory to get a working frequency close to the natural one and minimally affected by the feeding system. In this work, a systematic theoretical approach to reconstructing the resonator spectrum will be presented based on the edge length, the internal complexity, and the feeding system for microwave band-stop triangular Sierpinski resonators in the X-Band (8-12 GHz), using the equilateral triangle, and comparing the response of an electromagnetically simulated structure with the analytical results. Planar electromagnetic 2D simulations will be presented for silicon-based devices, i.e., metallized structures on high-resistivity silicon substrates with Sierpinski geometries, fed by microstrip lines. A perturbative approach will be used to predict the resonance frequencies of an internally subdivided triangle. Internally stretched triangles will also be presented, to propose a method to modulate the resonator's working frequency as an alternative to those characterised by increased internal complexity.

This paper presents a comprehensive review of the integration of fractal geometry in antenna engineering. It surveys key concepts, representative geometries, and major application domains of fractal antennas, arrays, and metasurfaces, highlighting their role in enabling multiband operation, miniaturization, enhanced directivity, and design flexibility in modern electromagnetic systems.

The integration of fractal geometry with electromagnetic theory has revolutionized antenna design, offering innovative solutions for modern telecommunications. ​ Fractal antennas leverage unique geometric properties such as self-similarity and space-filling to achieve multifrequency operation, miniaturization, high directivity antennas and arrays, and meta surfaces. ​ Practical applications include multi-frequency antennas. For example, geometries such as the Sierpinski monopole enable operation across multiple frequency bands, thereby reducing the need for separate antennas in applications including cellular communications, localization, and short-range wireless links.

In addition to supporting multiple resonances, certain fractal configurations—particularly space-filling curves like Hilbert and Koch—allow for significant miniaturization by lowering resonant frequencies without compromising performance. This capability is especially valuable in space-constrained scenarios such as automotive systems, portable devices, and embedded IoT platforms.

Beyond multiband operation and size reduction, fractal antennas can also support localized electromagnetic modes, often referred to as fractons and fractinos, which enhance directivity. By concentrating radiation in specific directions, these modes improve link reliability and signal strength in long-distance applications, including satellite communications.

Fractal concepts have also been successfully applied to metasurfaces, whose self-similar features confer distinct advantages over conventional designs. These include enabling multiband or wideband operation, achieving further size reduction, and providing a high degree of design flexibility for integration into complex system architectures.

As a result of these combined benefits—multiband capability, compactness, directivity control, and design versatility—fractal antenna engineering has been adopted in industries such as mobile communications and automotive, where high performance in compact and adaptable form factors is essential.

Chaotic behavior is a common feature of nonlinear dynamics, as well as
The subject of our discussion today is to study the stability of hyperchaos in high-dimensional systems. This study serves as an introductory guide to a discrete fractional four-dimensional hyperchaotic Rössler system with a Caputo-like operator, which is a complex system that can be used to study chaos in discrete fractional nonlinear dynamics. Our results demonstrate the existence of a hyperchaotic invariant set in these systems, leading to extended hyperchaotic transient behavior. The coexistence of chaos and hyperchaos is evident in the numerical results, which are presented as phase plots and bifurcation diagrams for various fractional orders and different parameters and initial conditions. These diagrams provide a comprehensive explanation of the dynamics of the proposed discrete system. This research substantiates the presence of chaos in discrete fractional hyperchaotic Rössler systems that are reminiscent of Caputo-like discrete systems. Control low is offered to display synchronization of coupled Caputo-like discrete fractional hyperchaotic Rössler systems and to force the states of the proposed system to converge asymptotically to zero. The findings of the study are demonstrated through the following numerical simulations, which have been a significant development in our research.

During near-ground flight, unmanned aerial vehicles (UAVs) experience enhanced aerodynamic efficiency due to the ground effect. However, the underlying flow mechanisms governing this optimal state—particularly the critical role of flow-structure orderliness—remain insufficiently clarified. To quantify flow orderliness and uncover its relationship with efficiency, this paper proposes a diagnostic method based on fractional-order entropy. The method utilizes unsteady lift time-series data obtained from computational fluid dynamics. Recognizing that this data embodies the historical memory and hereditary properties of the flow field, we introduce the Grünwald–Letnikov fractional-order derivative—well-suited for discrete sequences—to perform multi-scale fractional differentiation on the lift series. This process extracts embedded features reflecting the memory and scale-invariant characteristics of the flow structure. Permutation entropy is then calculated from the resulting fractional-order differential sequences and directly defined as the fractional-order entropy, which quantifies the dynamic orderliness of the flow field. Results indicate that at the flight altitude corresponding to peak aerodynamic efficiency, fractional-order entropy exhibits a distinct minimum. This extremum signifies that the flow field under this condition achieves the highest level of dynamic order, where ineffective turbulent dissipation is significantly suppressed and energy is efficiently converted into lift. Moreover, a strong monotonic correlation exists between fractional-order entropy and aerodynamic efficiency, confirming its effectiveness as a quantitative diagnostic indicator. From the novel perspective of flow-structure orderliness, this study systematically explains the formation mechanism of optimal efficiency in near-ground flight. The proposed fractional-order entropy framework not only provides a new theoretical tool for understanding complex ground effects but also lays a methodological foundation for the development of autonomous energy-saving flight control systems based on real-time flow-field perception.