Search Results (36)

This paper introduces a new class of Newton-type inequalities (NTIs) within the framework of extended fractional integral operators. This study begins by establishing a fundamental identity for generalized fractional Riemann–Liouville (FR-L) operators, which forms the basis for deriving various inequalities under different assumptions on the integrand. In particular, fractional counterparts of the classical $1/3$ and $3/8$ Simpson rules are obtained when the modulus of the first derivative is convex. The analysis is further extended to include functions that satisfy a Lipschitz condition or have bounded first derivatives. Moreover, an additional NTI is presented for functions of bounded variation, expressed in terms of their total variation. In all scenarios, the proposed results reduce to classical inequalities when the fractional parameters are specified accordingly, thus offering a unified perspective on numerical integration through fractional operators. Full article

In this paper, we investigate a class of fuzzy partially differentiable models for Caputo–Hadamard-type Goursat problems with generalized Hukuhara difference, which have been widely recognized as having a significant role in simulating and analyzing various kinds of processes in engineering and physical sciences. By transforming the fuzzy partially differentiable models into equivalent integral equations and employing classical Banach and Schauder fixed-point theorems, we establish the existence and uniqueness of solutions for the fuzzy partially differentiable models. Furthermore, in order to overcome the complexity of obtaining exact solutions of systems involving Caputo–Hadamard fractional derivatives, we explore numerical approximations based on trapezoidal and Simpson’s rules and propose three numerical examples to visually illustrate the main results presented in this paper. Full article

This article introduces the Third Space of Indian child welfare to theorize how Indigenous nations simultaneously engage and disrupt settler legal systems while building sovereign, care-based alternatives. Drawing from legal analysis, Indigenous political thought, and sociohistorical synthesis, I trace the historical continuity from boarding schools to today’s foster care removals, showing how child welfare operates as a colonial apparatus of family separation. In response, Native nations enact governance through three interrelated strategies: strategic legal engagement, kinship-based care, and tribally controlled family collectives. Building on Bruyneel’s theory of third space sovereignty, Simpson’s nested sovereignty, and Lightfoot’s global Indigenous rights framework, I conceptualize the Third Space as a dynamic field of Indigenous governance that transcends binary settler logics. These practices constitute sovereign abolitionist praxis. They reclaim kinship, resist carceral systems, and build collective futures beyond settler rule. Thus, rather than treating the Indian Child Welfare Act (ICWA) as a federal safeguard, I argue that tribes have repurposed ICWA as a legal and political vehicle for relational governance. This reframing challenges dominant crisis-based narratives and positions Indigenous child welfare as the center of a “global Indigenous politics of care” with implications for theories of sovereignty, family, and abolitionist futures across disciplines, geographies, and social groups. The article concludes by reflecting on the broader implications of the Third Space for other Indigenous and minoritized communities navigating state control and asserting self-determined care. Full article

In this study, we introduce a new hybrid identity that effectively combines Newton–Cotes and Gauss quadrature, allowing us to recover well-known formulas such as Simpson’s second rule and the left- and right-Radau two-point rules, among others. Building upon this flexible framework, we establish several new biparametrized fractal integral inequalities for functions whose local fractional derivatives are of a generalized convex type. In addition to employing tools from local fractional calculus, our approach utilizes the Hölder inequality, the power mean inequality, and a refined version of the latter. Further results are also derived using the concept of generalized concavity. To support our theoretical findings, we provide a graphical example that illustrates the validity of the obtained results, along with some practical applications that demonstrate their effectiveness. Full article

Emptying processes are operations frequently required in hydraulic installations by water utilities. These processes can result in drops to sub-atmospheric pressure pulses, which may lead to pipeline collapse depending on soil characteristics and the stiffness of a pipe class. One-dimensional mathematical models and 3D computational fluid dynamics (CFD) simulations have been employed to analyse the behaviour of the air–water interface during these events. The numerical resolution of these models is challenging, as 1D models necessitate solving a system of algebraic differential equations. At the same time, 3D CFD simulations can take months to complete depending on the characteristics of the pipeline. This presents a mathematical approach for directly solving air–water interactions in emptying processes involving entrapped air, providing a predictive tool for water utilities. The proposed mathematical approach enables water utilities to predict emptying operations in water pipelines without needing 2D/3D CFD simulations or the resolution of a differential algebraic equations system (1D model). A practical application is demonstrated in a case study of a 350 m long pipe with an internal diameter of 350 mm, investigating the influence of air pocket size, friction factor, polytropic coefficient, pipe diameter, resistance coefficient, and pipe slope. The mathematical approach is validated using an experimental facility that is 7.36 m long, comparing it with 1D mathematical models and 3D CFD simulations. The results confirm that the derived mathematical expression effectively predicts emptying operations in single water installations. Full article

The change grassland plant communities demonstrate with elevation has been one of the hot issues in ecological research, and there remain many unsolved problems. In order to further elucidate the rules of grassland plant community change with elevation, this study took the Burqin forest area as a research object, using field survey, redundancy analysis and grey correlation analysis to comprehensively assess the characteristics of change in grassland plant communities with elevation and the relationship of this evolution with environmental factors. The results showed that (1) the numbers of species, community biomass, community cover and community densities of grassland plant communities showed an “M” pattern with the increase in elevation. There were significant changes in the importance values and dominance of plants at different elevations; with increasing elevation, grassland plants became primarily dominated by cold-tolerant and well-adapted perennials. (2) The similarity coefficients of grassland plant communities at different elevations ranged from 0.06 to 0.62, i.e., from very dissimilar to moderately similar. (3) As the elevation increased, the Margalef species richness index, Shannon–Wiener diversity index, Simpson dominance index and Alatalo evenness index all showed an “M” pattern trend. (4) The degrees of correlation between temperature and precipitation and community biomass and species diversity were at a high level, and these were the most important environmental factors affecting the biomass and species diversity of grassland plant communities in the Burqin forest area. The results of this study can provide a theoretical basis for the rational utilization of grassland resources and for the sustainable development of grassland ecosystems in the Burqin forest area. Full article

Fractional calculus has been a concept used to acquire new variants of some well-known integral inequalities. In this study, our primary goal is to develop majorized fractional Simpson’s type estimates by employing a differentiable function. Practicing majorization theory, we formulate a new auxiliary identity by utilizing fractional integral operators. In order to obtain new bounds, we employ the idea of convex functions on the Niezgoda–Jensen–Mercer inequality for majorized tuples, along with some fundamental inequalities including the Hölder, power mean, and Young inequalities. Some applications to the quadrature rule and examples for special functions are provided as well. Interestingly, the main findings are the generalizations of many known results in the existing literature. Full article

Urban informality in developing economies like South Africa takes two forms: freestanding shacks are built in informal settlements, and backyard shacks are built in the yard of a formal house. The latter is evident in established townships around South African cities. In contrast to freestanding shacks, the number of backyard shacks has increased significantly in recent years. The study assessed the spatial patterns of backyard shacks in a formal settlement containing low-cost government houses (LCHs) using Unmanned Aerial Vehicle (UAV) products and landscape metrics. The backyard shacks were mapped using Object-Based Image Analysis (OBIA), which uses height information, vegetation index, and radiometric values. We assessed the effectiveness of rule-based and Random Forest (RF) OBIA techniques in detecting formal and informal structures. Informal structures were further classified as backyard shacks using spatial analysis. The spatial patterns of backyard shacks were assessed using eight shapes, aggregation, and landscape metrics. The analysis of the shape metrics shows that the backyard shacks are primarily square, as confirmed by a higher shape index value and a lower fractional dimension index value. The contiguity index of backyard shack patches is 0.6. The values of the shape metrics of backyard shacks were almost the same as those of formal and informal dwelling structures. The values of the assessed aggregation metrics of backyard shacks were more distinct from formal and informal structures compared with the shape metrics. The aggregation metrics show that the backyard shacks are less connected, less dense, and more isolated from each other compared with formal and freestanding shacks. The Shannon’s Diversity Index and Simpson’s Evenness Index values of informal settlements and formal areas with backyard shacks are almost the same. The results achieved in this study can be used to understand and manage informality in formal settlements. Full article

Invoking the matrix transfer technique, we propose a novel numerical scheme to solve the time-fractional advection–dispersion equation (ADE) with distributed-order Riesz-space fractional derivatives (FDs). The method adopts the midpoint rule to reformulate the distributed-order Riesz-space FDs by means of a second-order linear combination of Riesz-space FDs. Then, a central difference approximation is used side by side with the matrix transform technique for approximating the Riesz-space FDs. Based on this, the distributed-order time-fractional ADE is transformed into a time-fractional ordinary differential equation in the Caputo sense, which has an equivalent Volterra integral form. The Simpson method is used to discretize the weakly singular kernel of the resulting Volterra integral equation. Stability, convergence, and error analysis are presented. Finally, simulations are performed to substantiate the theoretical findings. Full article

In this work, a perturbed Milne’s quadrature rule for n-times differentiable functions with $L^p$-error estimates is derived. One of the most important advantages of our result is that it is verified for p-variation and Lipschitz functions. Several error estimates involving $L^p$-bounds are proven. These estimates are useful if the fourth derivative is unbounded in $L^{\infty }$-norm or the $L^p$-error estimate is less than the $L^{\infty }$-error estimate. Furthermore, since the classical Milne’s quadrature rule cannot be applied either when the fourth derivative is unbounded or does not exist, the proposed quadrature could be used alternatively. Numerical experiments showing that our proposed quadrature rule is better than the classical Milne rule for certain types of functions are also provided. The numerical experiments compare the accuracy of the proposed quadrature rule to the classical Milne rule when approximating different types of functions. The results show that, for certain types of functions, the proposed quadrature rule is more accurate than the classical Milne rule. Full article

Simpson’s rule is a numerical method used for approximating the definite integral of a function. In this paper, by utilizing mappings whose second derivatives are bounded, we acquire the upper and lower bounds for the Simpson-type inequalities by means of Riemann–Liouville fractional integral operators. We also study special cases of our main results. Furthermore, we give some examples with graphs to illustrate the main results. This study on fractional Simpson’s inequalities is the first paper in the literature as a method. Full article

Elevational gradients provide an excellent opportunity to assess biodiversity patterns and community structure. Previous studies mainly focus on higher elevations or are limited to small areas in mountainous regions. Little information can be found on amphibian biodiversity in middle- and low-elevational areas, hence our study was devoted to filling up the current gaps in these research areas. To understand the variability of biodiversity of amphibian species in the Fujian Junzifeng National Nature Reserve in eastern China, our study included taxonomic and phylogenetic components to describe the various patterns of regional and elevational distribution. The results showed that (1) most of the taxonomic and phylogenetic diversity metrics were correlated; with regard to the surveyed area, Faith’s phylogenetic diversity index (PD) and net relatedness index (NRI) were positively correlated with the Shannon–Wiener index (H’), Margalef index (DMG), and species richness (S), while negatively with the Pielou index; whereas for elevation, only the Pielou index was positively correlated with the nearest taxon index (NTI), but negatively with other indices; (2) taxonomic and phylogenetic diversities did not differ among the three survey locations but differed significantly along the elevational gradient; Simpson index, H’, S, and DMG had a hump-shaped relationship with elevations, and PD decreased gradually with the increase in elevation, whereas NRI and NTI sharply increased at the elevation above 900 m; (3) the species range size and the corresponding midpoint of amphibians were affected by a strong phylogenetic signal, which supports the elevational Rapoport’s rule upon removal of Pachytriton brevipes and Boulenophrys sanmingensis from the study. Full article

Integral inequalities are a powerful tool for estimating errors of quadrature formulas. In this study, some symmetric dual Simpson type integral inequalities for the classes of s-convex, bounded and Lipschitzian functions are proposed. The obtained results are based on a new identity and the use of some standard techniques such as Hölder as well as power mean inequalities. We give at the end some applications to the estimation of quadrature rules and to particular means. Full article

This paper focuses on computational technique to solve linear systems of Volterra integro-fractional differential equations (LSVIFDEs) in the Caputo sense for all fractional order $\mathrm{lins}\mathrm{in}\left(0,1\right]$ using two and three order block-by-block approach with explicit finite difference approximation. With this method, we aim to use an appropriate process to transform our problem into an analogous piecewise iterative linear algebraic system. Moreover, algorithms for treating LSVIFDEs using the above process have been developed, in order to express these solutions. In addition, numerical examples for our scheme are presented based on various kernels, including symmetry kernel and other sorts of separate kernels, are used to illustrate the validity, effectiveness and applicability of the suggested approach. Consequently, comparisons are performed with exact results using this technique, to communicate these approaches most general programs are written in Python V 3.8.8 software $\left(2021\right)$. Full article

The non-unique critical state represents the distance between the critical state line (CSL) and the isotropic consolidation line (ICL) that significantly varies with stress paths and particle size distribution of soils. A structural bounding surface plasticity model with spacing ratio r (SBSP-R model) was implemented using an explicit algorithm. However, the explicit algorithm did not well capture the non-unique critical state of soils with a large spacing ratio r, which prevented the soil mechanics research on non-unique critical state via finite element analysis. To overcome the limitation, the implicit algorithm of the SBSP-R model is formulated, and it mainly includes elastic prediction and plastic correction. The plastic correction is realized using the Newton–Simpson scheme with a controlling equation set related to consistency condition, plastic flow, hardening parameter, structural bounding surface, plastic modulus, and mapping rule. Case studies indicate that the implicit algorithm of the SBSP-R model is right and stable in predicting non-unique critical states. Comparisons between predicted and tested results indicate that the implicit algorithm of the SBSP-R model not only captures the critical state, stress-strain, and stress paths of various soils but also shows higher computational accuracy and efficiency compared with the previous explicit algorithm. These results indicate that the formulated implicit algorithm of the SBSP-R model is an alternative approach to the previous explicit algorithm. Full article