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Keywords = fractional-order method

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31 pages, 981 KB  
Review
Closing the Loop in Vitiviniculture and Spirit Production: From Grape Processing Waste to the Multifunctional Biochar
by Paraskevi Chalatsi-Diamanti, Ekavi Aikaterini Isari, Eleni Grilla, Petros Kokkinos and Ioannis K. Kalavrouziotis
Environments 2026, 13(7), 380; https://doi.org/10.3390/environments13070380 - 6 Jul 2026
Abstract
Viticulture, winemaking, and distillation industries produce substantial quantities of organic by-products (grape prunings, stems, marc-pomace, spent yeast, wine lees, distillation heads and tails, stillage), presenting both environmental challenges and promising opportunities for sustainable valorization. This narrative review offers a comprehensive analysis of the [...] Read more.
Viticulture, winemaking, and distillation industries produce substantial quantities of organic by-products (grape prunings, stems, marc-pomace, spent yeast, wine lees, distillation heads and tails, stillage), presenting both environmental challenges and promising opportunities for sustainable valorization. This narrative review offers a comprehensive analysis of the viticulture, winemaking, and distillation processes, along with the characterization of their solid and liquid by-products. Furthermore, it examines the chemical composition and mentions main valorization strategies for these residues, emphasizing their potential both as feedstocks for biochar production and as sources of valuable solvents. Additionally, various applications of biochars are explored, with particular focus on biochar derived from grape pruning and grape pomace-marc biomass, highlighting its effectiveness as a sorbent for environmental contaminants. Moreover, a broad range of biochar activation and modification methods that enhance physicochemical properties and expand functional applications is reviewed. Solvent-based biochar modification is also examined in order to introduce a novel concept: employing grape distillation liquid by-products (heads and tails fractions) to activate biochars made from grape prunings and pomace, creating a high-value sorbent while simultaneously valorizing two major waste streams. This dual-use strategy offers a new pathway for circular resource management within the viticulture and winemaking sectors. Full article
30 pages, 14689 KB  
Article
Fractional Texture-Guided and Boundary-Aware Perturbation Learning for Unsupervised Cross-Modality Medical Image Segmentation
by Xi Lin, Zhaoye Wu, Yu Wang, Haixiao Gong and Chenxi Huang
Fractal Fract. 2026, 10(7), 456; https://doi.org/10.3390/fractalfract10070456 - 6 Jul 2026
Abstract
Unsupervised domain adaptation (UDA) transfers knowledge from a labeled source domain to an unlabeled target domain and is particularly valuable in medical imaging, where dense annotations are costly and acquisition conditions vary. Cross-modality segmentation remains challenging because modality-dependent intensity and texture shifts alter [...] Read more.
Unsupervised domain adaptation (UDA) transfers knowledge from a labeled source domain to an unlabeled target domain and is particularly valuable in medical imaging, where dense annotations are costly and acquisition conditions vary. Cross-modality segmentation remains challenging because modality-dependent intensity and texture shifts alter image appearance, while teacher-generated pseudo-labels are often unreliable near anatomical boundaries. We propose a fractional texture-guided and boundary-aware perturbation-learning framework within a student–teacher scheme. On the source side, soft histogram transfer introduces target-related low-order intensity shifts. A multi-order fractional Gram discrepancy between shallow features of the intensity-transferred source and target images then provides a gradient signal for generating magnitude-normalized, range-clipped perturbations. This discrepancy is used as a perturbation cue rather than a direct alignment loss, exposing the student to target-relevant texture and edge-transition variation while preserving source annotations. On the target side, teacher logits are perturbed only within predicted boundary bands to model local contour uncertainty. Box-counting fractal boundary complexity guides the boundary-band width and logit perturbation scale and, together with predictive entropy, regulates pseudo-label supervision. Across five adaptation tasks, the proposed method achieves three-seed mean ± standard deviation Dice scores of 89.24 ± 0.12% and 82.01 ± 0.10% for cardiac MR→CT and CT→MR, 88.65 ± 0.29% and 90.43 ± 0.22% for abdominal MR→CT and CT→MR, and 84.76 ± 0.25% for bSSFP→LGE adaptation. Within the protocol-aware benchmark comparisons, the proposed method attains the highest average Dice score on four of the five tasks and is within 0.07 percentage points of the highest reported value on abdominal CT→MR. Ablation and operator-replacement studies further indicate that the source- and target-side pathways provide complementary benefits. Because all auxiliary perturbation and reliability-weighting modules are used only during adaptation, deployment requires only the adapted segmentation network, without additional inference-time modules or parameters. Full article
29 pages, 448 KB  
Article
Efficient Numerical Methods for Fractional- and Integer-Order Ordinary Differential Equations
by Marian Milev, Radan Miryanov and Yuri Dimitrov
Axioms 2026, 15(7), 501; https://doi.org/10.3390/axioms15070501 - 2 Jul 2026
Viewed by 197
Abstract
This paper proposes numerical methods for solving ordinary differential equations and fractional ordinary differential equations. The proposed methods are based on discretizations of first- and second-order derivatives, the L1 approximation of the Caputo fractional derivative, and a shifted L1-based approximation on a uniform [...] Read more.
This paper proposes numerical methods for solving ordinary differential equations and fractional ordinary differential equations. The proposed methods are based on discretizations of first- and second-order derivatives, the L1 approximation of the Caputo fractional derivative, and a shifted L1-based approximation on a uniform mesh. The discretizations of the integer-order derivatives depend on a free parameter, which enables the construction of numerical schemes with any prescribed order of accuracy in the interval (0,2] and supports the development of efficient, fast algorithms for computation of the solution. The discretizations of fractional derivatives employ the weights of the L1 approximation together with values of the Riemann zeta function. The convergence and accuracy of the numerical methods are analyzed theoretically. Numerical experiments confirm the theoretical results and demonstrate the improvement of the proposed methods over L1 schemes for the numerical solution of ordinary fractional differential equations. Full article
(This article belongs to the Special Issue Advances in Numerical Analysis and Its Applications)
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22 pages, 7695 KB  
Article
Prediction of Soil Salinity Parameters in the Songnen Plain Using FOD Processing and Machine Learning from Measured Hyperspectral Reflectance Under Different Surface Conditions
by Panpan Niu, Xingming Zheng, Weitong Zhao and Jianhua Ren
Remote Sens. 2026, 18(13), 2146; https://doi.org/10.3390/rs18132146 - 2 Jul 2026
Viewed by 210
Abstract
Soil salinization severely restricts ecosystem stability and the sustainable development of agricultural productivity. However, current understanding of the spectral–salinity quantitative relationships under the influence of surface cracking still remains limited. To address this gap, this study collected hyperspectral reflectance data (350–2500 nm) from [...] Read more.
Soil salinization severely restricts ecosystem stability and the sustainable development of agricultural productivity. However, current understanding of the spectral–salinity quantitative relationships under the influence of surface cracking still remains limited. To address this gap, this study collected hyperspectral reflectance data (350–2500 nm) from salt-affected soil in both cracked and uncracked surface conditions across the Songnen Plain, and applied fractional-order differentiation (FOD) processing with orders ranging from 0 to 2 and a step size of 0.1. Based on this, 14 types of FOD spectral indices were constructed, incorporating one-dimensional (1D), two-dimensional (2D), and three-dimensional (3D) structures. For each spectral index, the optimal fractional order and corresponding band combinations were first selected through Pearson correlation analysis for pH and EC under both surface conditions; subsequently, feature selection was performed using XGBoost-SHAP explainable analysis among the 14 optimal indices across different dimensions. Furthermore, the predictive performance of four modeling methods, including partial least squares regression (PLSR), Gaussian process regression (GPR), support vector regression (SVR), and random forest regression (RFR), was evaluated. The results showed that FOD transformations significantly enhanced correlations with EC and pH compared to raw reflectance. All prediction models demonstrated higher prediction accuracy under cracked surface conditions than uncracked surface conditions, indicating that desiccation cracks positively modulate spectral signals to enhance salinity information expression. Across different surface states, model performance generally followed the ranking: PLSR > GPR > SVR > RFR, with PLSR achieving the best predictions for EC and pH under cracked surfaces (R2 of 0.88 and 0.76, RMSE of 0.29 dS/m and 0.35). This study not only deepens the understanding of fractional-order spectral response mechanisms in saline–alkali soils but also provides methodological support for regional monitoring of soil salinization. Full article
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33 pages, 30969 KB  
Article
Adaptive Fractional Gradient Descent for Robust Deep Learning Optimization in Agricultural Pest Classification
by Nurullah Şahin, Davut Hanbay, Nuh Alpaslan and Mustafa İlçin
Appl. Sci. 2026, 16(13), 6611; https://doi.org/10.3390/app16136611 - 2 Jul 2026
Viewed by 178
Abstract
Agricultural pest infestations cause substantial global crop losses. Morphological similarities across species and structural variations across developmental stages render accurate identification a persistently expert-dependent and time-consuming process. Recent deep learning approaches have advanced automated pest classification; however, most efforts have concentrated on architectural [...] Read more.
Agricultural pest infestations cause substantial global crop losses. Morphological similarities across species and structural variations across developmental stages render accurate identification a persistently expert-dependent and time-consuming process. Recent deep learning approaches have advanced automated pest classification; however, most efforts have concentrated on architectural design, while optimization strategies have received comparatively little attention. This study proposes a novel optimization framework, hereafter referred to as Adaptive Fractional Gradient Descent (AFGD), that integrates the Grünwald–Letnikov (GL) fractional derivative into the backpropagation process of deep convolutional neural networks. Unlike standard gradient descent, the proposed method maintains a weighted history of past gradients. It dynamically adjusts the fractional order α via Bayesian optimization at regular training intervals, enabling the model to adaptively balance exploiting gradient memory against exploring new gradients throughout training. Experiments conducted on the IP102 benchmark dataset using DenseNet121, ResNet101, and EfficientNetB0 backbones demonstrated consistent accuracy improvements over standard gradient descent across all configurations. In the untrained setting, absolute test accuracy improved by 20.73, 11.51, and 11.01 percentage points for DenseNet121, ResNet101, and EfficientNetB0, although the absolute accuracy levels in this configuration remain modest. Under ImageNet pre-training, the proposed method yielded absolute gains of 6.69, 7.39, and 3.76 percentage points over the corresponding standard gradient baselines, with the highest absolute test accuracy of 70.81% recorded for DenseNet121. These findings indicate that fractional-order gradient control is a promising, architecturally complementary optimization strategy for robust pest classification, with broader implications for deep learning applications in precision agriculture. Full article
(This article belongs to the Special Issue Sustainable and Smart Agriculture)
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20 pages, 2700 KB  
Article
Numerical Investigation of Distributed-Order Cattaneo-Christov Model Based on Fractional Physics-Informed Neural Networks
by Xuehui Chen, Weijia Zhao, Jingbo Yang, Weidong Yang and Yang Liu
Fractal Fract. 2026, 10(7), 446; https://doi.org/10.3390/fractalfract10070446 - 29 Jun 2026
Viewed by 166
Abstract
A novel distributed-order Cattaneo–Christov model is proposed to effectively characterize non-classical heat conduction processes with memory effect and time–space relaxation behaviors originating from distributed-order fractional derivatives. A fractional physics-informed neural networks (fPINN) algorithm is employed to address both the forward and inverse problems [...] Read more.
A novel distributed-order Cattaneo–Christov model is proposed to effectively characterize non-classical heat conduction processes with memory effect and time–space relaxation behaviors originating from distributed-order fractional derivatives. A fractional physics-informed neural networks (fPINN) algorithm is employed to address both the forward and inverse problems of the distributed-order heat conduction model. For the forward problem, we propose an SfPINN algorithm that incorporates a squared loss term and employs an adaptive updating strategy for the loss-term weights. First, the boundary conditions are embedded into the network output such that they are automatically satisfied. In addition, we design a two-stage training strategy to enhance computational efficiency: in the first stage, the squared loss term associated with the initial condition is incorporated into the loss function; in the second stage, the squared residual term of the governing equation is introduced into the loss function. Numerical results show that the proposed algorithm outperforms the standard fPINN method in both solution accuracy and training iteration speed. For the inverse problem, the numerical results demonstrate that as the iteration number increases, the estimated parameter values progressively converge to their true values and finally stabilize. Full article
(This article belongs to the Special Issue Advanced Numerical Methods for Fractional Functional Models)
24 pages, 2624 KB  
Article
Simulation of Oxygen-Enriched Combustion Characteristics of Different Biomass Circulating Fluidized Beds Based on CPFD Model
by Yufeng Pei, Yuexin Wang, Xiuyan Zhang, Dandan Li, Nanhang Dong, Junhui Ma and Qing Wang
Processes 2026, 14(13), 2124; https://doi.org/10.3390/pr14132124 - 29 Jun 2026
Viewed by 160
Abstract
Biomass oxy-fuel combustion based on circulating fluidized bed (CFB) technology is one of the important pathways to achieving carbon neutrality due to its potential in carbon capture and negative carbon emissions. Combining biomass, as a substitute for coal, with oxy-fuel combustion technology can [...] Read more.
Biomass oxy-fuel combustion based on circulating fluidized bed (CFB) technology is one of the important pathways to achieving carbon neutrality due to its potential in carbon capture and negative carbon emissions. Combining biomass, as a substitute for coal, with oxy-fuel combustion technology can enrich CO2 while helping to control NOx emissions and carbon stock. In this study, a three-dimensional numerical model of a 20 t/h biomass CFB boiler was established based on the computational particle fluid dynamics (CPFD) method. Under an oxy-fuel atmosphere of 30% O2/65% CO2/5% H2O, the combustion characteristics of three typical biomass fuels—corn straw, rice husk, and poplar wood—were systematically compared, with emphasis on the furnace temperature distribution and the formation and emission of CO, NOX, and SO2. The results show that the axial temperature profiles all exhibit a rapid increase to a peak, followed by a gradual decrease. The peak temperatures in descending order are poplar wood (1091 K), corn straw (1084 K), and rice husk (1047 K), and the differences are mainly attributed to variations in volatile content, ash content, and calorific value. CO is primarily concentrated in the dense phase zone; it increases first and then decreases along the furnace height. CO generated from poplar wood combustion has the highest concentration at the furnace outlet, while the steady-state outlet mass fraction of NO is the lowest for poplar wood. Corn straw combustion yields the highest NO emission. Overall, the carbon stock of the three fuels is very low, and total CO emission is extremely low. NO concentration is jointly regulated by fuel nitrogen content and CO reduction, while SO2 emission is directly related to fuel sulfur content—corn straw and rice husk show significantly higher SO2 emission than poplar wood due to their higher sulfur content. In summary, fuel characteristics play a decisive role in the temperature field and pollutant formation during oxy-fuel combustion. This study provides a theoretical basis for the fuel selection and operational optimization of biomass oxy-fuel CFB boilers. Full article
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25 pages, 5475 KB  
Article
Robust Frequency Regulation of Hybrid Wind–PV Thermal Power Systems via Adaptive Fractional-Order PID Control
by Yevgeniy Muralev, Dinmukhambet Baimbetov, Samal Syrlybekkyzy, Mohamed Salem, Ali Bughneda and Khalid Yahya
Energies 2026, 19(13), 3076; https://doi.org/10.3390/en19133076 - 29 Jun 2026
Viewed by 254
Abstract
As modern electrical grids increasingly incorporate renewable generation—specifically from wind and solar–thermal installations—they face heightened volatility and operational complexities, which severely complicate load frequency regulation. While fractional-order proportional-integral-derivative (FOPID) controllers are commonly employed for this purpose, their conventional formulations rely on fixed fractional [...] Read more.
As modern electrical grids increasingly incorporate renewable generation—specifically from wind and solar–thermal installations—they face heightened volatility and operational complexities, which severely complicate load frequency regulation. While fractional-order proportional-integral-derivative (FOPID) controllers are commonly employed for this purpose, their conventional formulations rely on fixed fractional parameters that cannot adapt to fluctuating network conditions. To address this limitation, the present study develops an adaptive FOPID (AFOPID) control architecture capable of real-time adjustment of fractional orders, thereby enhancing regulatory effectiveness. The Coot Optimization Algorithm (COA) is utilized to optimally determine the operational parameters of all controllers under investigation. The proposed strategy is validated on a simulated hybrid power system comprising wind generation, solar–thermal units, and physical nonlinearities including governor dead band and generation rate constraints. A comparative analysis is conducted across four distinct operating scenarios, benchmarking the COA-tuned AFOPID against conventional PI, PID, and standard FOPID controllers. Quantitative results demonstrate that the proposed COA-AFOPID configuration achieves superior performance, with improvements in settling time up to 46.06% and reductions in ITAE index up to 89.89% compared to traditional methods. These findings confirm the enhanced stability and robustness of the proposed approach for frequency regulation in sustainable energy networks. Full article
(This article belongs to the Special Issue Energy Systems: Optimization, Modeling, and Simulation)
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34 pages, 1491 KB  
Article
Fractional Stochastic Modeling of Nonlinear Dynamical Systems: Application to an Electromechanical Process with Memory Effects
by Anwarud Din
Fractal Fract. 2026, 10(7), 440; https://doi.org/10.3390/fractalfract10070440 - 27 Jun 2026
Viewed by 200
Abstract
In this study, a comprehensive stochastic and fractional-order modeling framework is developed to investigate the dynamic behavior of a shunt DC motor under random disturbances and memory effects. The motor dynamics are formulated as a system of stochastic differential equations incorporating Gaussian noise [...] Read more.
In this study, a comprehensive stochastic and fractional-order modeling framework is developed to investigate the dynamic behavior of a shunt DC motor under random disturbances and memory effects. The motor dynamics are formulated as a system of stochastic differential equations incorporating Gaussian noise to represent uncertainties in the electrical and mechanical subsystems. The existence, stochastic ultimate boundedness, stationary distribution, and ergodic properties of the proposed model are established. To further enhance modeling capabilities, a modified Atangana–Baleanu–Caputo (mABC) fractional operator is introduced, enabling the incorporation of nonlocal memory effects inherent in electromechanical systems. The series solution is derived using the Laplace transform and the Adomian decomposition method to handle nonlinearities. Qualitative analysis of the solution is performed through fixed-point theory, while stability assessments utilize the T-Picard method. The results of the numerical simulation indicate that the stochastic model exhibits limited variability around the operating regimes, whereas the fractional-order representation is more effective at smoothing transient responses and limiting oscillatory behavior. The study proposes a realistic and adaptable method to analyze the dynamics of shunt DC motors with uncertainty and also presents useful information for the design and control of electromechanical systems. Full article
(This article belongs to the Section Life Science, Biophysics)
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31 pages, 3508 KB  
Article
Stability, Bifurcation Analysis and Chaos in a Discretized Fractional-Order Predator–Prey System with Nonlinear Functional Response
by Ibraheem M. Alsulami, Najat A. Alghamdi, M. T. Alharthi and Rizwan Ahmed
Mathematics 2026, 14(13), 2290; https://doi.org/10.3390/math14132290 - 27 Jun 2026
Viewed by 204
Abstract
This study examines a discrete fractional-order predator–prey system incorporating a Holling type-III functional response. The Caputo fractional derivative is employed because it naturally incorporates memory and hereditary effects while preserving biologically meaningful initial conditions. The system is formulated from a biologically relevant continuous [...] Read more.
This study examines a discrete fractional-order predator–prey system incorporating a Holling type-III functional response. The Caputo fractional derivative is employed because it naturally incorporates memory and hereditary effects while preserving biologically meaningful initial conditions. The system is formulated from a biologically relevant continuous fractional-order framework through the application of the piecewise constant argument approach, enabling an analysis of how memory-dependent effects and discrete dynamics influence predator–prey interactions. The existence and local stability of fixed points are determined by using the Jacobian matrix and eigenvalue conditions. The bifurcation of the positive fixed point is analyzed by using the center manifold and normal form methods. Numerical simulations, including bifurcation diagrams, phase portraits, and maximum Lyapunov exponent plots, confirm our analytical results and reveal periodic, quasiperiodic, and chaotic behavior. The findings of this study reveal that the combined influence of memory-dependent dynamics, nonlinear predator–prey interactions, and discrete-time effects can generate rich and complicated behaviors in fractional-order predator-prey systems. Full article
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31 pages, 3257 KB  
Article
OFOTD-FRMSST for LFM Signal Representation and Parameter Estimation Under Impulsive Noise
by Shan Zhang, Yong Guo and Lidong Yang
Fractal Fract. 2026, 10(7), 436; https://doi.org/10.3390/fractalfract10070436 - 26 Jun 2026
Viewed by 126
Abstract
Due to the memory and non-local characteristics of fractional calculus, fractional-order tracking differentiator (FOTD) performs excellently in suppressing impulse noise. However, the parameters of FOTD need to be manually adjusted according to the scene requirements, and cannot automatically maintain optimal performance in scenarios [...] Read more.
Due to the memory and non-local characteristics of fractional calculus, fractional-order tracking differentiator (FOTD) performs excellently in suppressing impulse noise. However, the parameters of FOTD need to be manually adjusted according to the scene requirements, and cannot automatically maintain optimal performance in scenarios where the signal and noise intensities change dynamically. To address this issue, this paper proposes a multi-parameter optimization-driven FOTD (OFOTD) based on envelope entropy, enhancing the adaptability of FOTD in complex scenarios. Furthermore, a fractional multisynchrosqueezing transform (FRMSST) is developed, and OFOTD-FRMSST is established to accurately represent the signal under impulsive noise. Finally, OFOTD-FRMSST is applied to parameter estimation of linear frequency modulation (LFM) signal, demonstrating its superiority in accuracy, noise robustness, and practicality. Experimental results demonstrate that, from both time domain and time-frequency plane, OFOTD achieves enhanced noise suppression performance through adaptive parameter optimization. Furthermore, in comparison with existing methods, OFOTD-FRMSST yields a more accurate signal representation under impulsive noise, thereby improving accuracy and noise robustness of parameter estimation. Full article
43 pages, 3040 KB  
Article
Hydrometeorological Disaster Insurance Modeling Based on Fractional Differential Equations for Climate Change Mitigation Within the Framework of SDG 13
by Hanifah Al Affiani, Muhamad Deni Johansyah, Endang Rusyaman, Sukono, Nurfadhlina Binti Abdul Halim, Alim Jaizul Wahid, Moch Panji Agung Saputra, Astrid Sulistya Azahra and Aceng Sambas
Mathematics 2026, 14(13), 2277; https://doi.org/10.3390/math14132277 - 26 Jun 2026
Viewed by 132
Abstract
Rainfall-index-based disaster insurance is an efficient approach to mitigating hydrometeorological losses. However, conventional premium pricing models generally assume memoryless stochastic dynamics that do not fully capture the long-range dependence inherent in rainfall data. This study develops a hydrometeorological disaster insurance model within a [...] Read more.
Rainfall-index-based disaster insurance is an efficient approach to mitigating hydrometeorological losses. However, conventional premium pricing models generally assume memoryless stochastic dynamics that do not fully capture the long-range dependence inherent in rainfall data. This study develops a hydrometeorological disaster insurance model within a fractional Black–Scholes framework to incorporate long-memory effects. The model is formulated using fractional differential equations and solved semi-analytically by integrating the Daftardar–Jafari Method (DJM) with the Kashuri–Fundo (KF) transform, yielding a closed-form solution expressed in terms of the Mittag–Leffler function. The proposed contract is structured as parametric rainfall insurance with a multi-layer payout mechanism based on percentiles corresponding to minor, moderate, and severe housing damage. The results show that variations in the fractional-order parameter significantly affect premium estimation. In particular, δ=0.5 recovers the classical model and tends to generate higher premiums than the fractional model with δ1=0.23153, whereas the model with δ2=0.73153 yields lower premiums. These findings indicate that fractional-order parameterization can accommodate diverse risk characteristics and policyholders’ economic capacities, enabling more adaptive, risk-sensitive premium structures. In line with SDG 13 (Climate Action), the proposed framework offers a climate-responsive disaster-mitigation strategy through accessible, actuarially relevant insurance design. Full article
(This article belongs to the Topic Fractional Calculus: Theory and Applications, 2nd Edition)
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16 pages, 5173 KB  
Article
Sol–Gel Synthesis and Characterization of Mullite–Spinel Ceramics Doped with Divalent (Co2+, Ni2+) Transition Metal Ions
by Tsvetan Dimitrov, Rositsa Titorenkova, Ivan Tsanev, Daniela Kovacheva, Mariela Minova and Irena Markovska
Crystals 2026, 16(7), 413; https://doi.org/10.3390/cryst16070413 - 25 Jun 2026
Viewed by 234
Abstract
Co- and Ni-doped mullite–spinel ceramics were synthesized via a sol–gel method followed by high-temperature sintering in order to investigate the influence of dopant type on the phase evolution, microstructure, and optical properties. X-ray diffraction analysis confirmed the formation of a multiphase system consisting [...] Read more.
Co- and Ni-doped mullite–spinel ceramics were synthesized via a sol–gel method followed by high-temperature sintering in order to investigate the influence of dopant type on the phase evolution, microstructure, and optical properties. X-ray diffraction analysis confirmed the formation of a multiphase system consisting of mullite and spinel phases, with a residual amorphous fraction, the amount of which decreases with increasing temperature. FTIR and Raman spectroscopy indicate progressive structural ordering of both spinel and aluminosilicate networks during thermal treatment, with differences in crystallization behavior between Co- and Ni-containing system. UV–Vis spectroscopy revealed characteristic absorption bands arising from d–d electronic transitions of Co2+ and Ni2+ ions in the ceramic matrix, reflecting differences in their local coordination environments and optical behavior. Colorimetric analysis showed that Co-doped samples exhibit intense blue coloration, whereas Ni-doped ceramics display greenish-blue hues. The temperature-dependent evolution of the L*, a*, and b* parameters correlate with structural changes. The results suggest that the type of additive influences the phase evolution and optical response in mullite–spinel ceramics, in agreement with structural and spectroscopic analyses. Full article
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32 pages, 2491 KB  
Article
A Spectral-fPINN Framework for Fractional Optimal Control Problems
by Yonis Gulzar and Ishtiaq Ali
Computation 2026, 14(7), 146; https://doi.org/10.3390/computation14070146 - 25 Jun 2026
Viewed by 198
Abstract
Fractional optimal control problems provide an effective mathematical framework for modeling dynamical systems with memory, hereditary behavior, and anomalous diffusion effects. However, the nonlocal nature of Caputo fractional operators and the reduced regularity of fractional solutions pose significant challenges for the development of [...] Read more.
Fractional optimal control problems provide an effective mathematical framework for modeling dynamical systems with memory, hereditary behavior, and anomalous diffusion effects. However, the nonlocal nature of Caputo fractional operators and the reduced regularity of fractional solutions pose significant challenges for the development of accurate and efficient computational methods. In this paper, we develop a spectral-fractional Physics-Informed Neural Network (Spectral-fPINN) framework for solving fractional optimal control problems governed by Caputo fractional differential equations. The proposed methodology combines normalized shifted Legendre spectral approximations, fractional operational matrix formulations, and physics-informed optimization within a unified computational framework. Unlike conventional PINN and fPINN approaches, which directly approximate the unknown solution variables, the proposed framework predicts the spectral coefficient vectors associated with the shifted Legendre basis functions, yielding a low-dimensional global representation with improved approximation efficiency. Caputo fractional derivatives are evaluated through spectral operational matrices, while the resulting optimization problem is discretized using Gauss–Legendre quadrature and solved through gradient-based optimization. In addition, a theoretical analysis of the proposed Spectral-fPINN framework is presented, including approximation, consistency, stability, and convergence results, together with error estimates and residual control properties. Several benchmark linear and nonlinear fractional optimal control problems are investigated to validate the proposed methodology. The numerical results demonstrate excellent agreement with exact solutions, very small residual errors, and rapid spectral coefficient decay, confirming the high-order accuracy and robustness of the proposed approach. Overall, the proposed Spectral-fPINN framework provides an accurate, stable, and computationally efficient methodology for solving a broad class of fractional optimal control problems. Full article
(This article belongs to the Special Issue Nonlinear System Modelling and Control—2nd Edition)
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8 pages, 1016 KB  
Proceeding Paper
Impact of Recent Precipitation Trends on the Performance of Rooftop Rainwater Harvesting Systems: A Storage Yield Assessment for Mediterranean Urban Conditions
by Tuğçe Başar and Şahnaz Tiğrek
Environ. Earth Sci. Proc. 2026, 44(1), 31; https://doi.org/10.3390/eesp2026044031 - 24 Jun 2026
Viewed by 102
Abstract
Rooftop rainwater harvesting (RWH) offers a practical adaptation option for Mediterranean cities where water scarcity is amplified by seasonal rainfall and climate variability. This study reports early findings from a simplified monthly water balance screening model for a typical residential building, driven by [...] Read more.
Rooftop rainwater harvesting (RWH) offers a practical adaptation option for Mediterranean cities where water scarcity is amplified by seasonal rainfall and climate variability. This study reports early findings from a simplified monthly water balance screening model for a typical residential building, driven by ERA5-Land monthly precipitation for Antalya and İzmir (Türkiye). Scenarios cover roof areas of 250–3000 m2 and practical tank capacities of 2–100 m3 under a fixed non-potable demand of 0.20 m3/day. The model tracks monthly storage dynamics and supply demand in order to compute demand coverage and monthly reliability (i.e., fraction of months in which full demand is met). Reliability-based storage thresholds (≥0.80) are derived for four evaluation windows (1996–2010, 2011–2025, 1996–2025, 1950–2025) to explore climate sensitivity. In parallel, a guideline-style sizing which is consistent with the Turkish rainwater harvesting guideline is implemented using a three-day storage rule based on the wettest month potential. To enable a like-for-like comparison, the collection losses are harmonized by setting loss to 0.10 in the simulation and efficiency to 0.90 in the guideline method. The results show stable thresholds for Antalya but stronger period sensitivity in İzmir. They also quantify cases where guideline sizing does not achieve the target reliability under dry season constraints. This approach supports the rapid, climate-aware pre-design of small- to medium-scale urban RWH systems. Full article
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