Search Results (66)

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

In this survey, we have included the recent results on Lyapunov-type inequalities for differential equations of fractional order associated with Dirichlet, nonlocal, multi-point, anti-periodic, and discrete boundary conditions. Our results involve a variety of fractional derivatives such as Riemann–Liouville, Caputo, Hilfer–Hadamard, $\psi$-Riemann–Liouville, Atangana–Baleanu, tempered, half-linear, and discrete fractional derivatives. Full article

This study investigates fuzzy fixed points and fuzzy best proximity points for fuzzy mappings within the newly introduced framework $\theta$-fuzzy metric spaces that extends various existing fuzzy metric spaces. We establish novel fixed-point and best proximity-point theorems for both single-valued and multivalued mappings, thereby broadening the scope of fuzzy analysis. Furthermerefore, we have for aore, we apply one of our key results to derive conditions, ensuring the existence and uniqueness of a solution to Hadamard $\Psi$-Caputo tempered fuzzy fractional differential equations, particularly in the context of the SIR dynamics model. These theoretical advancements are expected to open new avenues for research in fuzzy fixed-point theory and its applications to hybrid models within $\theta$-fuzzy metric spaces. Full article

Mathematics and physics are deeply interconnected. In fact, physics relies on mathematical tools like calculus and differential equations. The aim of this article is to introduce tempered Riemann–Liouville (RL) fractional operators and their properties with applications in mathematical physics. The tempered RL substantial fractional derivative is also defined, and its properties are discussed. The Laplace transforms (LTs) of the introduced fractional operators are evaluated. The Hyers–Ulam stability and the existence of a novel tempered fractional differential equation are examined. Moreover, a fractional integro-differential kinetic equation is formulated, and the LT is used to find its solution. A growth model and its graphical representation are established, highlighting the role of novel fractional operators in modeling complex dynamical systems. The developed mathematical framework offers valuable insights into solving a range of scenarios in mathematical physics. Full article

Chlorophyll fluorescence (ChlF) parameters serve as non-destructive indicators of vegetation photosynthetic function and are widely used as key input parameters in photosynthesis–fluorescence models. The rapid acquisition of the spatiotemporal dynamics of ChlF parameters is crucial for enhancing remote sensing applications and improving carbon cycle modeling. While hyperspectral reflectance offers a promising data source for estimating ChlF parameters, previous studies have relied primarily on spectral indices derived from specific datasets, which often lack robustness. In this study, we simultaneously monitored ChlF parameters and spectral reflectance in leaves from different species, growth stages, and canopy positions within a temperate deciduous forest. We developed a data-driven partial least squares regression (PLSR) model by integrating fractional-order derivative (FOD) spectral transformation with multiple feature selection methods to predict ChlF parameters. The results demonstrated that FOD spectra effectively improved prediction accuracy compared to conventional PLSR attempts. Among the feature selection algorithms, the least absolute shrinkage and selection operator (LASSO) and stepwise regression (Stepwise) methods outperformed others. Furthermore, the LASSO-based PLSR model that used low-order (<1) FOD spectra achieved high predictive performance for NPQ (R² = 0.60, RPD = 1.60, NRMSE = 0.16), ΦP (R² = 0.73, RPD = 1.94, NRMSE = 0.11), ΦN (R² = 0.62, RPD = 1.62, NRMSE = 0.12), and ΦF (R² = 0.54, RPD = 1.48, NRMSE = 0.15). These findings suggest that the integration of FOD spectral transformation and appropriate feature selection enables the simultaneous estimation of multiple ChlF parameters, providing valuable insights for the retrieval of ChlF parameters from hyperspectral data. Full article

In this paper, we focus on the existence of positive solutions for a class of p-Laplacian tempered fractional diffusion equations involving a lower tempered integral operator and a Riemann–Stieltjes integral boundary condition. By introducing certain new local growth conditions and establishing an a priori estimate for the Green’s function, several sufficient conditions on the existence of positive solutions for the equation are derived by using a fixed point theorem. Interesting points are that the tempered fractional diffusion equation contains a lower tempered integral operator and that the boundary condition involves the Riemann–Stieltjes integral, which can be a changing-sign measure. Full article

The importance of this research comes from the several applications of the Mathieu equation and its generalizations in many scientific fields. Two models of fractional Mathieu equations are provided using Katugampola fractional derivatives in the sense of Riemann-Liouville and Caputo. Each model contains two fractional derivatives with unique fractional orders, periodic forcing of the cosine stiffness coefficient, and many extensions and generalizations. The Banach contraction principle is used to prove that each model under consideration has a unique solution. Our results are applied to four real-life problems: the nonlinear Mathieu equation for parametric damping and the Duffing oscillator, the quadratically damped Mathieu equation, the fractional Mathieu equation’s transition curves, and the tempered fractional model of the linearly damped ion motion with an octopole. Full article

This study treats the problem of Finite Time Stability Analysis (FTSA) and Finite Time Feedback Control (FTFC), using a Linear Matrix Inequalities Approach (LMIA). It specifically focuses on Takagi–Sugeno fuzzy Time Delay Fractional-Order Systems (TDFOS) that include nonlinear perturbations and interval Time Varying Delays (ITVDs). We consider the case of the Caputo Tempered Fractional Derivative (CTFD), which generalizes the Caputo Fractional Derivative (CFD). Two main results are presented: a two-step procedure is provided, followed by a more relaxed single-step procedure. Two examples are presented to show the reduction in conservatism achieved by the proposed methods. The first is a numerical example, while the second involves the FTFC of an inverted pendulum, which exhibits both symmetrical dynamics for small angular displacements and asymmetrical dynamics for larger deviations. Full article

The method of the equivalent system of fractional integral equations is used to prove the existence results of a unique solution for initial value problems corresponding to various classes of nonlinear fractional differential equations involving the tempered $\Psi$–Caputo fractional derivative. These include equations with their right side depending on ordinary as well as fractional-order derivatives, or fractional integrals of the solution. Full article

Ecballium elaterium, also known as squirting cucumber, is a plant which is widespread in temperate regions of Europe, Africa and Asia. The plant is considered to be one of the oldest used drugs. In the last decades, E. elaterium has been widely studied as a source of triterpene metabolites named cucurbitacins, often found as glycosylated derivatives, used by the plant as defensive agents. Such metabolites exhibit several biological activities, including cytotoxic, anti-inflammatory, and anti-cancer. Interestingly, the bioactive properties of E. elaterium extracts have been investigated in dozens of studies, especially by testing the apolar fractions, including the essential oils, extracted from leaves and fruits. The purpose of this review is to provide an overview of the chemical profile of different parts of the plants (leaves, flowers, and seeds) analyzing the methods used for structure elucidation and identification of single metabolites. The pharmacological studies on the isolated compounds are also reported, to highlight their potential as good candidates for drug discovery. Full article

As claimed in many papers, the equivalence between the Caputo-type fractional differential problem and the corresponding integral forms may fail outside the spaces of absolutely continuous functions, even in Hölder spaces. To avoid such an equivalence problem, we define a “new” appropriate fractional integral operator, which is the right inverse of the Caputo derivative on some Hölder spaces of critical orders less than 1. A series of illustrative examples and counter-examples substantiate the necessity of our research. As an application, we use our method to discuss the BVP for the Langevin fractional differential equation $\frac{d_{\psi }^{\beta ,\mu }}{d{t}^{\beta }}\left(\frac{d_{\psi }^{\alpha ,\mu }}{d{t}^{\alpha }}+\lambda \right)x\left(t\right)=f(t,x\left(t\right)),t\in [a,b],\lambda \in \mathbb{R}$, for $f\in C\left([a,b]\times \mathbb{R}\right)$ and some critical orders $\beta ,\alpha \in (0,1)$, combined with appropriate initial or boundary conditions, and with general classes of $\psi$-tempered Hilfer problems with $\psi$-tempered fractional derivatives. The BVP for fractional differential problems of the Bagley–Torvik type was also studied. Full article

This paper aims to demonstrate that, beyond the small world of Riemann–Liouville and Caputo derivatives, there is a vast and rich world with many derivatives suitable for specific problems and various theoretical frameworks to develop, corresponding to different paths taken. The notions of time and scale sequences are introduced, and general associated basic derivatives, namely, right/stretching and left/shrinking, are defined. A general framework for fractional derivative definitions is reviewed and applied to obtain both known and new fractional-order derivatives. Several fractional derivatives are considered, mainly Liouville, Hadamard, Euler, bilinear, tempered, q-derivative, and Hahn. Full article

This paper considers a nonlinear fractional-order boundary value problem ${\hspace{0.25em}}^H{\mathbb{D}}_{a,g}^{{\alpha }_1,\beta ,\mu }x\left(t\right)+f(t,x\left(t\right){,}^H{\mathbb{D}}_{a,g}^{{\alpha }_2,\beta ,\mu }x\left(t\right))=0,$ for $t\in [a,b]$, ${\alpha }_1\in (1,2]$, ${\alpha }_2\in (0,1]$, $\beta \in [0,1]$ with appropriate integral boundary conditions on the Hölder spaces. Here, f is a real-valued function that satisfies the Hölder condition, and ${ }^H{\mathbb{D}}_{a,g}^{\alpha ,\beta ,\mu }$ represents the tempered-Hilfer fractional derivative of order $\alpha >0$ with parameter $\mu \in {\mathbb{R}}^{+}$ and type $\beta \in [0,1]$. The corresponding integral problem is introduced in the study of this issue. This paper addresses a fundamental issue in the field, namely the circumstances under which differential and integral problems are equivalent. This approach enables the study of differential problems using integral operators. In order to achieve this, tempered fractional calculus and the equivalence problem of the studied problems are introduced and studied. The selection of an appropriate function space is of fundamental importance. This paper investigates the applicability of these operators on Hölder spaces and provides a comprehensive rationale for this choice. Full article

In this paper, we focus on the existence of positive solutions for a singular tempered sub-diffusion fractional model involving a quasi-homogeneous nonlinear operator. By using the spectrum theory and computing the fixed point index, some new sufficient conditions for the existence of positive solutions are derived. It is worth pointing out that the nonlinearity of the equation contains a tempered fractional sub-diffusion term, and is allowed to possess strong singularities in time and space variables. In particular, the quasi-homogeneous operator is a nonlinear and non-symmetrical operator. Full article

In this paper, the tempered $\mathsf{\Psi }$–Riemann–Liouville fractional derivative and the tempered $\mathsf{\Psi }$–Caputo fractional derivative of order $n-1<\alpha <n\in \mathbb{N}$ are introduced for $C^{n-1}$–functions. A nonlinear version of the second Henry–Gronwall inequality for integral inequalities with the tempered $\mathsf{\Psi }$–Hilfer fractional integral is derived. By using this inequality, an existence and uniqueness result and a sufficient condition for the non-existence of blow-up solutions of nonlinear tempered $\mathsf{\Psi }$–Caputo fractional differential equations are proved. Illustrative examples are given. Full article