Search Results (2,079)

Asymptotic solutions are derived to model the development of atmospheric internal gravity waves generated by latent heating in a two-dimensional configuration involving a vertically-sheared background flow. The mathematical model comprises nonlinear partial differential equations derived from the conservation laws of fluid dynamics under the anelastic approximation where the background density and temperature vary with altitude. The latent heating is represented by a horizontally-periodic but vertically-localized nonhomogeneous forcing term in the energy conservation equation. This generates gravity waves that are considered as perturbations to the background flow and are expressed as perturbation series, with the leading-order contributions being the solutions of linearized equations. Taking into account the nonlinear terms at the next order gives expressions for the effects of the waves on the background mean flow. Due to the vertical shear, there is a critical level where momentum and energy are transferred from the wave modes to the mean flow. The asymptotic solutions show that the wave–mean-flow interaction is nonlocal and occurs over the range of altitudes from the thermal forcing level up the critical level. This is in contrast to what occurs in the case of waves forced by an oscillatory lower boundary, where the interaction is typically localized around the critical level. It is found that the wave drag is negative above the thermal forcing level, making the mean flow velocity more negative, but it becomes positive as the waves approach the critical level, indicating wave absorption in this region. There is wave transmission through the critical level, as well as absorption, and the extent of transmission depends on the depth of the latent heating profile. The mean potential temperature is reduced above the thermal forcing level and enhanced at the critical level, a situation that could ultimately lead to the development of convective instabilities. Full article

This paper presents a symmetric positive definite (SPD) coupling between the boundary element method (BEM) and the finite element method (FEM) in the framework of the numerical manifold method (NMM) for two-dimensional linear elastic static problems. The BEM subdomain is treated as a single mathematical patch whose local approximation is derived from the displacement boundary integral equation, thereby preserving the nonlocal nature of BEM. The remaining domain is covered by a finite element mesh, with each node defining a patch and the associated shape functions serving as weight functions. Weight functions are defined over the entire mathematical cover, with explicit zero values outside the support of each patch. This global definition ensures that the partition of unity holds everywhere and enables the global displacement approximation to be expressed as a superposition of contributions from all patches. Within this unified framework, the interface between the BEM and FEM subdomains emerges naturally as a transition zone of weight functions, rather than a distinct boundary. Displacement continuity is automatically satisfied through the partition of unity, and traction equilibrium is approximately enforced through the variational formulation. To fully incorporate the coupling formulation into the minimum potential energy framework, the tractions on the BEM patch are eliminated in favor of displacements using the displacement boundary integral equation (BIE). Prescribed tractions on the BEM patch are enforced via a penalty method. The resulting algebraic system is symmetric by construction and remains positive definite when either constant or isoparametric boundary elements are used. This work serves as a proof-of-concept study for the SPD coupling framework with constant elements. Numerical examples demonstrate the accuracy and convergence of the method. The results show that the coupling procedure preserves the intrinsic convergence properties of each subdomain: the BEM part converges at a rate close to unity for displacements and approximately 2.0 for stresses, while the FEM part achieves quadratic convergence for both. The study also reveals that near-singular integrals in the strain BIE can affect the convergence rate when the element size becomes sufficiently small. Full article

The Mittag-Leffler function holds significant importance in fractional calculus due to its extensive applications in addressing challenges across science, engineering, biology, hydrology, and earth sciences. Notably, the closed-form solution of a time-fractional model naturally emerges as the Mittag-Leffler function (MLF), necessitating precise and efficient computations. Consequently, numerical approximations are essential for accurately calculating the Mittag-Leffler function. In this study, we develop a straightforward yet precise real pole rational approximation for the Mittag-Leffler function. We demonstrate first-order convergence and L-acceptability, which aid in mitigating unwanted oscillations. Additionally, we create an effective and precise first-order generalized exponential time differencing scheme to solve the time-fractional reaction–diffusion equations. We obtain and prove the convergence result using Grönwall-type inequality. Several numerical experiments are conducted to confirm the efficiency and accuracy of the proposed numerical scheme compared with exact solutions. The computational efficiency of the proposed method is compared with another existing first-order numerical technique. Furthermore, our proposed scheme is crucial for developing higher-order predictor–corrector schemes for solving time-fractional models. Full article

The endeavor to refute hidden variable theories underlying quantum theory has yielded the discipline of contextual sets. A plethora of various kinds of sets of arbitrary structure in any dimension have been developed, alongside extensive experimental validation. These advancements incited us to investigate to what extent we might move beyond hidden variables, engineer contextual sets, and find their applications within quantum theory itself, without any reference to hidden variable models. To this end, we consider possible applications of contextual sets in quantum computation, cryptography, pseudo-telepathy, and nonlocality, as well as generating them from error-correction protocols, complex Hadamard gates, and simple quantum gates. We found that the results in this field are still scarce, and we therefore investigated the directions in which future research might be carried out and the potential obstacles to realizing such undertakings in the past. Full article

Recently, a great amount of attention has been focused on the study of fractional and nonlocal operators of the elliptic type both for pure mathematical research and in view of concrete real-world applications. We are interested in proving the existence and nonexistence of solutions of a minimizing problem involving a fractional Laplacian with weight. We consider the nonlocal minimizing problem on ${\mathbb{H}}_0^s\left(\Omega \right)\subset L^{q_s}\left(\Omega \right)$, with $q_s:={\displaystyle \frac{2n}{n-2s}}$, $s\in (0,1)$, and $n\ge 3$${\displaystyle \underset{\begin{array}{c}u\in {\mathbb{H}}_0^s\left(\Omega \right)\\ {\left|\right|u\left|\right|}_{L^{q_s}\left(\Omega \right)}=1\end{array}}{inf}{\int }_{{\mathbb{R}}^n}{p\left(x\right)|(-\Delta )}^{\frac{s}{2}}{u\left(x\right)|}^{2}dx-\lambda {\int }_{\Omega }{\left|u\left(x\right)\right|}^{2}dx,}$ where $\Omega$ is a bounded domain in ${\mathbb{R}}^n,p:{\mathbb{R}}^n\to \mathbb{R}$ is a given positive weight presenting a global positive minimum $p_0>0$ at $a\in \Omega$, and $\lambda$ is a real constant. The objective of this paper is to show that minimizers do not exist for some $k,s,\lambda$, and n. After that, we show some nonexistence results thanks to a fractional Pohozaev identity and fractional Hardy inequality. Full article

This paper investigates non-fractional operators, a type of nonlocal operator, within the framework of Orlicz spaces. Using inclusions between certain function spaces, we prove the continuity and/or compactness of generalized operators in Orlicz spaces and show that solutions exist for integral equations of fractional order. We also introduce a generalized Hilfer-type derivative and examine the equivalence of differential and integral problems. Finally, we relate these results to the study of compositional p-Laplacian fractional problems involving generalized Hilfer fractional derivatives. Among other things, we prove the existence of solutions to such problems in Orlicz and Orlicz–Sobolev spaces. Full article

The purpose of this instrumental case study, employing both qualitative and quantitative data, was to investigate how novice teachers from non-local and urban areas used community assets and local funds of knowledge (FoK) in their STEM instruction in a remote Montana town. While non-local teachers often make up a large share of many rural communities’ teaching workforce, those teachers might lack the social, cultural, and community knowledge that they need to teach with place-conscious approaches. Therefore, this study explored how “new-to-town” teachers, with limited personal ties to a community, learn about their rural community and how they apply this knowledge to their teaching context. Additionally, this study examined which research-established factors that improve rural STEM education were deemed most important for novice, rural teachers. The exploration employed a floodlight research approach, whereby a census of the authentic pedagogical actions of the subjects was documented rather than investigating the efficacy of a single method. Data sources included qualitative instruments like concept maps and semi-structured interviews, alongside quantitative measures like ranked best-practices data and place-conscious lesson ratios, to provide both depth of interpretation and breadth of comparison across participants. Results from the deductive thematic analysis suggest that novice teachers aspire to implement asset-based pedagogical approaches in STEM instruction and possess some methods for integration but struggle to learn of local community assets without modeling and mentorship. Additionally, an unexpected pattern emerged from the findings: Novice, newcomer teachers that employed place-conscious lessons were more likely to remain teaching in their position. While this association cannot be interpreted causally, it might suggest that place-conscious mentorship practices may play a role in improving instruction and support the retention of non-local teachers in rural communities however, further, more robust exploration is warranted of this exploratory finding. Findings from this study can be used to inform recommendations for school districts, post-secondary institutions, and rural communities on how best to support beginning rural teachers with limited community connections. Full article

This study proposes an improved retinal blood vessel segmentation method to enhance the diagnosis of microvascular retinal complications. The proposed method extracts local shape features from retinal images utilizing a fractional Hessian matrix, which models blood vessels as surface structures characterized by ridges and valleys resulting from variations in curvature. The methodology integrates adaptive principal curvature estimation with a new framework leveraging the fractional Hessian matrix with nonsingular and nonlocal kernels. The effectiveness of the suggested method is assessed using publicly accessible datasets, including DRIVE, HRF, STARE, and some real images obtained from a local hospital. The proposed segmentation achieves 96.77% accuracy and 98.82% specificity on the DRIVE database, 96.91% accuracy and 98.69% specificity on STARE, and 95.90% accuracy and 98.36% specificity on the HRF database. Optimal parameters for the fractional order and Gaussian standard deviation were empirically determined by maximizing segmentation accuracy. Our findings show that the proposed approach achieves competitive performance compared to the listed methods, including several deep learning approaches, while maintaining significant computational efficiency. The output of the suggested method can be further utilized with deep learning techniques, which will be applied in the clinical context of diabetic retinopathy and glaucoma to identify abnormalities likely related to disease progression and different stages. Full article

We identify a fundamental tension between general relativity and spectral geometry arising from the global, nonlocal character of spectral data versus the local causal dynamics of spacetime. To resolve this, we postulate spectral invariance, $\delta {\Lambda }_n=0$, requiring the eigenvalues of the Laplace–Beltrami operator to remain fixed under physical evolution. This condition yields a compensatory relation between metric deformations and eigenfunction amplitudes, suggesting a kinematic coupling linking energy distribution to spacetime curvature. From the second variation of the associated energy functional, we derive a rank-4 tensor proportional to the inverse DeWitt supermetric on superspace and a contracted rank-2 tensor proportional to the spacetime metric, and we recover a invariance law of the energy functional in configuration space. Spectral invariance may suggest a framework in which geometry and energy are co-defined through fixed spectral data. Full article

In recent years, image denoising has seen a shift from traditional non-local self-similarity methods like BM3D to deep-learning based approaches that use learnable convolutions and attention mechanisms. While pixel-level attention is effective at capturing long-range relationships similar to non-local self-similarity based methods, it incurs extremely high computational costs that scale quadratically with image resolution. As an alternative, channel-wise attention is resolution-independent and computationally efficient but may miss crucial spatial details. In this paper, an adjustable attention mechanism is introduced that bridges the gap between pixel and channel attentions. In the proposed model, average pooling and variable-size convolutions are added before attention calculation to adjust spatial resolution and, thus, allow dynamical adjustment of computational complexity. This adjustable attention is applied in a transformer-based U-Net architecture and achieves performance comparable to state-of-the-art methods in both real and Gaussian blind denoising tasks. To be more concrete, the proposed method achieves a Peak Signal-to-Noise Ratio of 39.65 dB and a Structural Similarity Index Measure of 0.913 on the Smartphone Image Denoising Dataset. Therefore, the proposed method demonstrates a balance between efficiency and denoising quality. Full article

Time–space fractional diffusion equations are widely used to model anomalous transport in heterogeneous biological tissues, where memory effects, spatial nonlocality, and coefficient variability are intrinsically coupled. However, existing numerical approaches typically treat these aspects in isolation, and a fully discrete framework that simultaneously accounts for heterogeneity, long-memory effects, and computational efficiency remains lacking. In this work, a fully discrete numerical method is developed and analyzed. The method integrates heterogeneous diffusion coefficients and memory-efficient temporal discretization within a unified variational framework. It combines a finite element approximation of a spectral fractional elliptic operator with an implicit $L^1$ discretization of the Caputo derivative enhanced by a sum-of-exponentials approximation of the memory kernel. Unconditional stability, preservation of a discrete energy structure, and a fully discrete error estimate are established, explicitly separating temporal, spatial, and kernel approximation errors. The proposed approach reduces memory complexity from $\mathcal{O}\left(N\right)$ to $\mathcal{O}(logN)$ without compromising accuracy. Numerical experiments confirm the theoretical convergence rates, demonstrate stable behavior across all tested configurations, and illustrate the impact of heterogeneous coefficients on anomalous transport dynamics. Full article

In this investigation, the nonlinear dust-acoustic waves in the lunar terminator region are studied in a three-component complex plasma comprising Boltzmann-distributed electrons and ions and inertial, cold, negatively charged dust grains. The fluid model is reduced, via the reductive perturbation technique, to a planar Korteweg–de Vries (KdV) equation that governs the evolution of small-amplitude dust-acoustic structures in this environment. Hirota’s direct method is then employed to derive exact multiple-soliton solutions, which allow us to examine the parameter dependence of dust-acoustic solitons and to characterize their overtaking collisions. The analysis shows that the soliton polarity and amplitude are controlled by the equilibrium electron–ion density ratio and the electron-to-ion temperature ratio, and that multi-soliton interactions remain elastic, with only finite phase shifts after collision. In the second part of the study, the planar integer KdV model is generalized to a time-fractional KdV (FKdV) equation to incorporate nonlocal temporal memory effects in the dust-acoustic dynamics. This FKdV equation is analyzed using two analytical approximation schemes: the Tantawy technique, recently proposed as a direct and rapidly convergent approach to fractional evolution equations, and the new iterative method, a widely used high-accuracy scheme in the fractional literature. For both methods, higher-order approximations are constructed, and their absolute and global maximum residual errors are quantified. The results demonstrate that the Tantawy technique provides compact approximations with superior accuracy and stability compared with the new iterative method for the present FKdV-soliton problem. The combined integer- and fractional-analytic framework provides a physically transparent framework for understanding how nonlinearity, dispersion, and fractional memory jointly shape dust-acoustic solitary structures in the electrostatically complex lunar terminator plasma, which is of paramount interest for future lunar missions like Luna-25 and Luna-27. Full article

This article investigates the numerical solution of the two-dimensional Poisson equation defined over a rectangular domain subject to a double integral nonlocal boundary condition. We propose a finite difference scheme by discretizing the integral term using the two-dimensional trapezoidal rule. The main difficulty of this problem is that, in the non-classical case, we cannot use the method of separation of variables and decompose the problem into one-dimensional problems. Our approach involves reducing the integral boundary condition from the complete domain to the interior points and strategically partitioning the computational domain into the boundary and interior points. We propose a method that allows us to find a solution by solving the Poisson equation with classical boundary conditions, and using the solutions found to construct a solution to a problem with a nonlocal integral condition. This method requires solving a linear system whose dimension is much smaller than the original. Under certain conditions on the kernel, the proposed method is correct. Full article

We develop a fast numerical method for solving nonlinear diffusion equations with memory phenomena, a class of problems arising within viscoelastic materials, anomalous transport, and hereditary systems. The primary computational problem is the nonlocal temporal dependence captured by Volterra-type memory operators, which makes direct evaluation scale quadratically with the number of time steps ($O\left(N_t^2\right)$), rendering prolonged simulations prohibitively expensive. To address this bottleneck, we develop a novel synthesis that combines a high-order spectral method for spatial discretization with a fast memory algorithm based on a sum-of-exponentials approximation. The spectral method obtains exponential spatial convergence for smooth solutions. At the same time, the fast memory algorithm reduces memory usage and computational complexity to $O\left(N_t\right)$, yielding computational speedups exceeding 414x for prolonged simulations. We rigorously prove that the proposed scheme preserves the discrete energy dissipation law of the continuous system under mild assumptions on the memory kernel, thereby ensuring unconditional stability. Error analysis verifies spectral accuracy in space and first-order temporal convergence. Extensive numerical experiments using exponentially decaying and weakly singular kernels validate the theoretical results and illustrate the method’s effectiveness for modeling viscoelastic transport phenomena and irregular diffusion in complex systems. Full article

This paper develops an operator-reduction framework for a class of coupled implicit Caputo-Hadamard fractional differential systems subject to nonlocal Hadamard integral constraints. The system, involving fractional derivatives in both state and auxiliary variables, is resolved through a pointwise contraction argument that eliminates the auxiliary components and reduces the problem to a two-dimensional fixed-point operator acting on a Banach space of continuous functions. This reduction overcomes the compactness obstruction that arises in direct multi-component formulations. Under explicit growth and smallness conditions, the existence of at least one solution is established via Mönch’s fixed-point theorem. By imposing strengthened Lipschitz hypotheses, the reduced operator becomes a strict contraction on an invariant ball, yielding uniqueness and Ulam-Hyers stability with explicit constant $C_{UH}=1/(1-\Lambda )$. A fully computed example demonstrates the verifiability of the theoretical assumptions and illustrates how the smallness condition $\Lambda <1$ governs both existence and stability. The results establish a systematic operator-based approach for implicit Caputo-Hadamard systems with nonlocal integral constraints. Full article