Search Results (1,569)

Asymmetric functional-order (variable-order) fractional diffusion–wave equations (FO-FDWEs) introduce considerable computational challenges, as the fractional order of the derivatives can vary spatially or temporally. To overcome these challenges, a novel spectral method employing generalized fractional-order Chelyshkov wavelets (FO-CWs) is developed to efficiently solve such equations. In this approach, the Riemann–Liouville fractional integral operator of variable order is evaluated in closed form via a regularized incomplete Beta function, enabling the transformation of the governing equation into a system of algebraic equations. This wavelet-based spectral scheme attains extremely high accuracy, yielding significantly lower errors than existing numerical techniques. In particular, numerical results show that the proposed method achieves notably improved accuracy compared to existing methods under the same number of basis functions. Its strong convergence properties allow high precision to be achieved with relatively few wavelet basis functions, leading to efficient computations. The method’s accuracy and efficiency are demonstrated on several practical diffusion–wave examples, indicating its suitability for real-world applications. Furthermore, it readily applies to a wide class of fractional partial differential equations (FPDEs) with spatially or temporally varying order, demonstrating versatility for diverse applications. Full article

This paper introduces a novel fractional-order model using the Caputo derivative operator to investigate the virus dynamics of adaptive immune responses. Two infection routes, namely cell-to-cell and virus-to-cell transmissions, are incorporated into the dynamics. Our research establishes the existence and uniqueness of positive and bounded solutions through the application of the generalized mean-value theorem and Banach fixed-point theory methods. The fractional-order model is shown to be Ulam–Hyers stable, ensuring the model’s resilience to small errors. By employing the normalized forward sensitivity method, we identify critical parameters that profoundly influence the transmission dynamics of the fractional-order virus model. Additionally, the framework of optimal control theory is used to explore the characterization of optimal adaptive immune responses, encompassing antibodies and cytotoxic T lymphocytes (CTL). To assess the influence of memory effects, we utilize the generalized forward–backward sweep technique to simulate the fractional-order virus dynamics. This study contributes to the existing body of knowledge by providing insights into how the interaction between virus-to-cell and cell-to-cell dynamics within the host is affected by memory effects in the presence of optimal control, reinforcing the invaluable synergy between fractional calculus and optimal control theory in modeling within-host virus dynamics, and paving the way for potential control strategies rooted in adaptive immunity and fractional-order modeling. Full article

Urban expansion fragments once-contiguous forest patches, generating pronounced edge gradients that modulate soil physicochemical properties and biodiversity. We quantified how fragmentation reshaped the soil microbiome continuum and its implications for soil carbon storage in a temperate urban mixed deciduous forest. A total of 18 plots were considered in this study, with six plots for each fragment type. Intact interior forest (F), internal forest path fragment (IF), and external forest path fragment (EF) soils were sampled at 0–15, 15–30, and 30–45 cm depths and profiled through phospholipid-derived fatty acid (PLFA) chemotyping and amino sugar proxies for living microbiome and microbial-derived necromass assessment, respectively. Carbon fractionation was performed through the chemical oxidation method. Diversity indices (Shannon–Wiener, Pielou evenness, Margalef richness, and Simpson dominance) were calculated based on the determined fatty acids derived from the phospholipid fraction. The microbial biomass ranged from 85.1 to 214.6 nmol g⁻¹ dry soil, with the surface layers of F exhibiting the highest values (p < 0.01). Shannon diversity declined systematically from F > IF > EF. The microbial necromass varied from 11.3 to 23.2 g⋅kg⁻¹. Fragmentation intensified the stratification of carbon pools, with organic carbon decreasing by approximately 14% from F to EF. Our results show that EFs possess a declining microbiome continuum that weakens their carbon sequestration capacity in urban forests. Full article

With the rapid proliferation of real-time digital communication, particularly in multimedia applications, securing transmitted image data has become a vital concern. While chaotic systems have shown strong potential for cryptographic use, most existing approaches rely on low-dimensional, integer-order architectures, limiting their complexity and resistance to attacks. Advances in fractional calculus and memristive technologies offer new avenues for enhancing security through more complex and tunable dynamics. However, the practical deployment of high-dimensional fractional-order memristive chaotic systems in hardware remains underexplored. This study addresses this gap by presenting a secure image transmission system implemented on a field-programmable gate array (FPGA) using a universal high-dimensional memristive chaotic topology with arbitrary-order dynamics. The design leverages four- and five-dimensional hyperchaotic oscillators, analyzed through bifurcation diagrams and Lyapunov exponents. To enable efficient hardware realization, the chaotic dynamics are approximated using the explicit fractional-order Runge–Kutta (EFORK) method with the Caputo fractional derivative, implemented in VHDL. Deployed on the Xilinx Artix-7 AC701 platform, synchronized master–slave chaotic generators drive a multi-stage stream cipher. This encryption process supports both RGB and grayscale images. Evaluation shows strong cryptographic properties: correlation of $-6.1081\times {10}^{-5}$, entropy of 7.9991, NPCR of 99.9776%, UACI of 33.4154%, and a key space of $2^{1344}$, confirming high security and robustness. Full article

This study addresses the practical stability analysis of Riemann-Liouville fractional-order nonlinear systems with time delays. We first establish a rigorous formulation of initial conditions that aligns with the properties of Riemann-Liouville fractional derivatives. Subsequently, a generalized definition of practical stability is introduced, specifically tailored to accommodate the hybrid dynamics of fractional calculus and time-delay phenomena. By constructing appropriate Lyapunov-Krasovskii functionals and employing an enhanced Razumikhin-type technique, we derive sufficient conditions ensuring practical stability in the $L_p$-norm sense. The theoretical findings are validated through illustrative example for fractional order nonlinear systems with time delays. Full article

In this study, we focus on solving the nonlinear time-fractional Hirota–Satsuma coupled Korteweg–de Vries (KdV) and modified Korteweg–de Vries (MKdV) equations, using the Yang transform iterative method (YTIM). This method combines the Yang transform with a new iterative scheme to construct reliable and efficient solutions. Readers can understand the procedures clearly, since the implementation of Yang transform directly transforms fractional derivative sections into algebraic terms in the given problems. The new iterative scheme is applied to generate series solutions for the provided problems. The fractional derivatives are considered in the Caputo sense. To validate the proposed approach, two numerical examples are analysed and compared with exact solutions, as well as with the results obtained from the fractional reduced differential transform method (FRDTM) and the q-homotopy analysis transform method (q-HATM). The comparisons, presented through both tables and graphical illustrations, confirm the enhanced accuracy and reliability of the proposed method. Moreover, the effect of varying the fractional order is explored, demonstrating convergence of the solution as the order approaches an integer value. Importantly, the time-fractional Hirota–Satsuma coupled KdV and modified Korteweg–de Vries (MKdV) equations investigated in this work are not only of theoretical and computational interest but also possess significant implications for achieving global sustainability goals. Specifically, these equations contribute to the Sustainable Development Goal (SDG) “Life Below Water” by offering advanced modelling capabilities for understanding wave propagation and ocean dynamics, thus supporting marine ecosystem research and management. It is also relevant to SDG “Climate Action” as it aids in the simulation of environmental phenomena crucial to climate change analysis and mitigation. Additionally, the development and application of innovative mathematical modelling techniques align with “Industry, Innovation, and Infrastructure” promoting advanced computational tools for use in ocean engineering, environmental monitoring, and other infrastructure-related domains. Therefore, the proposed method not only advances mathematical and numerical analysis but also fosters interdisciplinary contributions toward sustainable development. Full article

Normal and anomalous diffusion processes are characterized by the time evolution of the mean square displacement of a diffusing molecule ${\sigma }^2\left(t\right)$. When ${\sigma }^2\left(t\right)$ is a power function of time, the process is described by a fractional subdiffusion, fractional superdiffusion or normal diffusion equation. However, for other forms of ${\sigma }^2\left(t\right)$, diffusion equations are often not defined. We show that to describe diffusion characterized by ${\sigma }^2\left(t\right)$, the g-subdiffusion equation with the fractional Caputo derivative with respect to a function g can be used. Choosing an appropriate function g, we obtain Green’s function for this equation, which generates the assumed ${\sigma }^2\left(t\right)$. A method for solving such an equation, based on the Laplace transform with respect to the function g, is also described. Full article

In this work, we consider a generalized form of the classical $(2+1)$-dimensional sine-Gordon system. The mathematical model considers a generalized reaction term, and the two-dimensional Laplacian includes the presence of space-fractional derivatives of the Riesz type with two different differentiation orders in general. The system is equipped with a conserved quantity that resembles the energy functional in the integer-order scenario. We propose a numerical model to approximate the solutions of the fractional sine-Gordon equation. A discretized form of the energy-like quantity is proposed, and we prove that it is conserved throughout the discrete time. Moreover, the analysis of consistency, stability, and convergence is rigorously carried out. The numerical model is implemented computationally, and some computer simulations are presented in this work. As a consequence of our simulations, we show that the discrete energy is approximately conserved throughout time, which coincides with the theoretical results. Full article

Some studies performed in our laboratory on pyrrole and its derivatives pointed towards the enrichment of the evaluations of these promising chemical structures for the potential treatment of neurodegenerative conditions in general and Parkinson’s disease in particular. A classical Paal-Knorr cyclization approach is applied to synthesize the basic hydrazine used for the formation of the designed series of hydrazones (15a–15g). The potential neurotoxic and neuroprotective effects of the newly synthesized derivatives were investigated in vitro using different models of induced oxidative stress at three subcellular levels (rat brain synaptosomes, mitochondria, and microsomes). The results identified as the least neurotoxic molecules, 15a, 15d, and 15f applied at a concentration of 100 µM to the isolated fractions. In addition, the highest statistically significant neuroprotection was observed for 15a and 15d at a concentration of 100 µM using three different injury models on subcellular fractions, including 6-hydroxydopamine in rat brain synaptosomes, tert-butyl hydroperoxide in brain mitochondria, and non-enzyme-induced lipid peroxidation in brain microsomes. The hMAOA/MAOB inhibitory activity of the new compounds was studied at a concentration of 1 µM. The lack of a statistically significant hMAOA inhibitory effect was observed for all tested compounds, except for 15f, which showed 40% inhibitory activity. The most prominent statistically significant hMAOB inhibitory effect was determined for 15a, 15d, and 15f, comparable to that of selegiline. The corresponding selectivity index defined 15f as a non-selective MAO inhibitor and all other new hydrazones as selective hMAOB inhibitors, with 15d indicating the highest selectivity index of >471. The most active and least toxic representative (15d) was evaluated in vivo on Rotenone based model of Parkinson’s disease. The results revealed no microscopically visible alterations in the ganglion and glial cells in the animals treated with rotenone in combination with 15d. Full article

In this paper, we first introduce a parametric identity for generalized differentiable functions using a generalized fractal–fractional integral operators. Based on this identity, we establish several variants of parameterized inequalities for functions whose local fractional derivatives in absolute value satisfy generalized convexity conditions. Furthermore, we demonstrate that our main results reduce to well-known Ostrowski- and Simpson-type inequalities by selecting suitable parameters. These inequalities contribute to finding tight bounds for various integrals over fractal spaces. By comparing the classical Hölder and Power mean inequalities with their new generalized versions, we show that the improved forms yield sharper and more refined upper bounds. In particular, we illustrate that the generalizations of Hölder and Power mean inequalities provide better results when applied to fractal integrals, with their tighter bounds supported by graphical representations. Finally, a series of applications are discussed, including generalized special means, generalized probability density functions and generalized quadrature formulas, which highlight the practical significance of the proposed results in fractal analysis. Full article

Vaccine adjuvants are non-immunogenic agents that enhance or modulate immune responses to co-administered antigens and are essential to modern vaccines. Despite their importance, few are approved for human use. The rise of new pathogens and limited efficacy of some existing vaccines underscore the need for more advanced and effective formulations, particularly for vulnerable populations. Aluminum-based adjuvants are commonly used in vaccines and effectively promote humoral immunity. However, they mainly induce a Th2-biased response, making them suboptimal for diseases requiring cell-mediated immunity. In contrast, saponin-based adjuvants from the Quillajaceae family elicit a more balanced Th1/Th2 response and generate antigen-specific cytotoxic T cells (CTL). Due to ecological damage and limited availability caused by overharvesting Quillaja saponaria Molina barks, efforts have intensified to identify alternative plant-derived saponins with enhanced efficacy and lower toxicity. Quillaja brasiliensis (A.St.-Hil. and Tul.) Mart. (syn. Quillaja lancifolia D.Don), a related species native to South America, is considered a promising renewable source of Quillajaceae saponins. In this review, we highlight recent advances in vaccine adjuvant research, with a particular focus on saponins extracted from Q. brasiliensis leaves as a sustainable alternative to Q. saponaria saponins. These saponin fractions are structurally and functionally comparable, exhibiting similar adjuvant activity when they were formulated with different viral antigens. An alternative application involves formulating saponins into nanoparticles known as ISCOMs (immune-stimulating complexes) or ISCOM-matrices. These formulations significantly reduce hemolytic activity while preserving strong immunoadjuvant properties. Therefore, research advances using saponin-based adjuvants (SBA) derived from Q. brasiliensis and their incorporation into new vaccine platforms may represent a viable and sustainable solution for the development of more less reactogenic, safer, and effective vaccines, especially for diseases that require a robust cellular immunity. Full article

In this article, we investigate the fractional soliton solutions as well as the chaotic analysis of the fourth-order nonlinear Ablowitz–Kaup–Newell–Segur wave equation. This model is considered an intriguing high-order nonlinear partial differential equation that integrates additional spatial and dispersive effects to extend the concepts to more intricate wave dynamics, relevant in engineering and science for understanding complex phenomena. To examine the solitary wave solutions of the proposed model, we employ sophisticated analytical techniques, including the generalized projective Riccati equation method, the new improved generalized exponential rational function method, and the modified F-expansion method, along with mathematical simulations, to obtain a deeper insight into wave propagation. To explore desirable soliton solutions, the nonlinear partial differential equation is converted into its respective ordinary differential equations by wave transforms utilizing $\beta$-fractional derivatives. Further, the solutions in the forms of bright, dark, singular, combined, and complex solitons are secured. Various physical parameter values and arrangements are employed to investigate the soliton solutions of the system. Variations in parameter values result in specific behaviors of the solutions, which we illustrate via various types of visualizations. Additionally, a key aspect of this research involves analyzing the chaotic behavior of the governing model. A perturbed version of the system is derived and then analyzed using chaos detection techniques such as power spectrum analysis, Poincaré return maps, and basin attractor visualization. The study of nonlinear dynamics reveals the system’s sensitivity to initial conditions and its dependence on time-decay effects. This indicates that the system exhibits chaotic behavior under perturbations, where even minor variations in the starting conditions can lead to drastically different outcomes as time progresses. Such behavior underscores the complexity and unpredictability inherent in the system, highlighting the importance of understanding its chaotic dynamics. This study evaluates the effectiveness of currently employed methodologies and elucidates the specific behaviors of the system’s nonlinear dynamics, thus providing new insights into the field of high-dimensional nonlinear scientific wave phenomena. The results demonstrate the effectiveness and versatility of the approach used to address complex nonlinear partial differential equations. Full article

The chiral nonlinear Schrödinger equation (CNLSE) serves as a simplified model for characterizing edge states in the fractional quantum Hall effect. In this paper, we leverage the generalization and parameter inversion capabilities of physics-informed neural networks (PINNs) to investigate both forward and inverse problems of 1D and 2D CNLSEs. Specifically, a hybrid optimization strategy incorporating exponential learning rate decay is proposed to reconstruct data-driven solutions, including bright soliton for the 1D case and bright, dark soliton as well as periodic solutions for the 2D case. Moreover, we conduct a comprehensive discussion on varying parameter configurations derived from the equations and their corresponding solutions to evaluate the adaptability of the PINNs framework. The effects of residual points, network architectures, and weight settings are additionally examined. For the inverse problems, the coefficients of 1D and 2D CNLSEs are successfully identified using soliton solution data, and several factors that can impact the robustness of the proposed model, such as noise interference, time range, and observation moment are explored as well. Numerical experiments highlight the remarkable efficacy of PINNs in solution reconstruction and coefficient identification while revealing that observational noise exerts a more pronounced influence on accuracy compared to boundary perturbations. Our research offers new insights into simulating dynamics and discovering parameters of nonlinear chiral systems with deep learning. Full article

This article discusses the transformation of a continuous-time model of the fractional system into a discrete-time model of the fractional system. For the continuous-time model, the exact solution of the nonlinear equation with fractional derivatives (FDs) that has the form of the damped rotator type with power non-locality in time is obtained.This equation with two FDs and periodic kicks is solved in the general case for the arbitrary orders of FDs without any approximations. A three-stage method for solving a nonlinear equation with two FDs and deriving discrete maps with memory (DMMs) is proposed. The exact solutions of the nonlinear equation with two FDs are obtained for arbitrary values of the orders of these derivatives. In this article, the orders of two FDs are not related to each other, unlike in previous works. The exact solution of nonlinear equation with two FDs of different orders and periodic kicks are proposed. Using this exact solution, we derive DMMs that describe a kicked damped rotator with power-law non-localities in time. For the discrete-time model, these damped DMMs are described by the exact solution of nonlinear equations with FDs at discrete time points as the functions of all past discrete moments of time. An example of the application, the exact solution and DMMs are proposed for the economic growth model with two-parameter power-law memory and price kicks. It should be emphasized that the manuscript proposes exact analytical solutions to nonlinear equations with FDs, which are derived without any approximations. Therefore, it does not require any numerical proofs, justifications, or numerical validation. The proposed method gives exact analytical solutions, where approximations are not used at all. Full article

The production of spirits generates significant amounts of waste in the form of fusel oils-previously treated mainly as an environmental problem. This paper presents an innovative installation designed to recover valuable components from this difficult waste. The key achievement is the effective separation and recovery of pyrazine derivatives-natural aromatic compounds with high utility value in the food, cosmetics and pharmaceutical industries. The designed system allows for the recovery of as much as 98% of pyrazines and isoamyl alcohol and isobutanol fractions with a purity above 96%, which is a significant advance compared to previous disposal methods. The installation was designed to be consistent with the idea of a circular economy, maximizing the use of by-products and minimizing losses. The results of the work indicate that fusel oils, previously perceived as waste, can become a source of valuable secondary raw materials, and the presented solution opens up new possibilities for the sustainable development of the alcohol industry. Full article