JANOLI International Journal of Mathematical Science
The JANOLI International Journal of Mathematical Science (JIJMS) is a prestigious, peer-reviewed journal dedicated to the dissemination of high-quality research in the field of mathematical sciences. The journal provides a global platform for mathematicians, researchers, and academics to publish original research, reviews, and case studies that contribute to the advancement of mathematical knowledge and its applications across various scientific and engineering disciplines. Subject: Mathematics Sub Domains: Mathematical Modelling Real Analysis Statistical Methods Numerical Methods Functional Analysis Topology Operations Research Cosmology Crptology Modern Algebra Discrete Mathematics
Shanti Educational Research Foundation, India
Casson Blood in Narrow Stenosed Arteries under MHD and Slip Shear Dependent Viscosity Effects
This study investigates the hemodynamic behavior of Casson blood flow through a stenosed arterial segment under the combined effects of magnetic field (MHD), wall slip, and shear-dependent viscosity. The mathematical model incorporates a steady, axisymmetric, and incompressible flow with a transverse magnetic field applied to an electrically conducting Casson fluid. Analytical solutions are obtained under mild stenosis and low magnetic Reynolds number approximations. The influence of Hartmann number, slip parameter, and yield stress on velocity distribution, volumetric flow rate, and wall shear stress is analyzed. Results reveal that increasing the magnetic field strength significantly suppresses the volumetric flow rate while enhancing wall shear stress due to intensified Lorentz forces. Conversely, higher wall slip reduces the flow resistance and shear stress, promoting smoother motion. The findings provide important insights into the magnetohydrodynamic regulation of blood flow in diseased arteries and the potential therapeutic relevance of wall slip effects in microvascular transport.
Casson fluid, Magnetohydrodynamics (MHD), Wall shear stress, Slip parameter, Stenosed artery, Hartmann number, Hemodynamics, Non-Newtonian blood flow.
A review on MHD flow and heat transfer of Casson fluid
The present study examines the impact of the Lorentz force on the flow and heat transfer characteristics of a Casson fluid over a stretching sheet embedded in a non-Darcy porous medium, incorporating a slip boundary condition. The governing partial differential equations (PDEs) are transformed into ordinary differential equations (ODEs) using suitable similarity transformations and are subsequently solved numerically. The effects of key parameters including the magnetic parameter, porosity parameter, inertia parameter, and slip parameter on the velocity and temperature profiles, skin friction coefficient, and local Nusselt number are presented through tables. The results reveal that an increase in the slip parameter enhances the skin friction coefficient but reduces the heat transfer rate at the sheet surface. Furthermore, an increase in the Casson parameter elevates the fluid temperature while diminishing the velocity profile.
MHD, Casson Fluid (CF), Slip Boundary Condition (SBC), Darcy-Forchheimer Model (DFM)
Hydrodynamic Bearings for Space Mechanisms Considering Surface Roughness and Elastic Deformation
The Space Tribology Handbook makes only a brief mention of hydrodynamic bearings, stating that “hydrodynamic bearings are not considered, as such bearings are generally either impractical for space applications or do not involve tribological behavior.” While this assessment holds true for most space mechanisms, certain specialized applications do employ hydrodynamic bearings. One notable example is pumps used in spacecraft thermal fluid loops, where such bearings enable complete rotor encapsulation and eliminate the need for dynamic seals between the rotor and stator. Some heritage designs of space-qualified pumps exist and are discussed as background in this paper. However, the existing literature lacks comprehensive guidelines for the sizing and design of mechanisms incorporating hydrodynamic bearings. The work conducted by nano SPACE and ESTL aims to address this gap through dedicated activities in bearing design, modeling, and experimental validation. In space mechanisms, hydrodynamic bearings are vital for ensuring precise, low-friction, and long-life rotational motion under microgravity and vacuum conditions. This study investigates the influence of surface roughness and structural deformation (elastic or thermoelastic) on the performance characteristics of hydrodynamic bearings operating in space environments. The analysis incorporates the modified Reynolds equation, accounting for roughness parameters, film thickness variations, and bearing elastic compliance. Results indicate that micro-level roughness significantly alters pressure distribution and load capacity, while elastic deformation plays a key role in maintaining film stability under varying thermal and dynamic loads. The findings are crucial for designing reliable, lightweight, and durable bearings for satellites, gyroscopes, reaction wheels, and robotic joints.
Hydrodynamic bearings; Space tribology; Microgravity lubrication; Thermal fluid loop pumps; Modified Reynolds equation; Surface roughness effects
Influence of Transverse Roughness and Porous Medium on Conjugate Heat Transfer in an Inclined Nanofluid-Filled Cavity: A Lattice Boltzmann Study
This study numerically investigates the influence of transverse wall roughness and porous medium characteristics on conjugate heat transfer within an inclined cavity filled with Al₂O₃/water nanofluid using the Lattice Boltzmann Method (LBM). The physical model consists of a two-dimensional inclined enclosure with roughened heated walls, where solid–fluid heat conduction coupling is considered to account for conjugate effects. The porous structure is modeled using the Darcy–Forchheimer formulation, while nanofluid thermophysical properties are evaluated based on effective medium theory. The governing equations for momentum and energy transport are solved using a double-distribution LBM scheme. The effects of key dimensionless parameters—including Rayleigh number (10³ ≤ Ra ≤ 10⁶), Darcy number (10⁻⁵ ≤ Da ≤ 10⁻²), porosity (0.4 ≤ ε ≤ 0.9), roughness amplitude and pitch, nanoparticle volume fraction (0 ≤ φ ≤ 0.05), and cavity inclination angle (0° ≤ γ ≤ 90°)—are systematically examined. Results reveal that transverse roughness significantly enhances local mixing and disrupts thermal boundary layers, leading to noticeable augmentation in average Nusselt number, particularly at higher Rayleigh numbers. However, excessive roughness amplitude induces localized recirculation zones that may suppress global heat transfer under low Darcy number conditions. Increasing porosity and Darcy number promotes convective dominance, while higher nanoparticle volume fractions improve effective thermal conductivity but slightly damp fluid motion due to viscosity enhancement. The inclination angle plays a crucial role in reorienting buoyancy-driven circulation, producing optimal thermal performance at intermediate angles. The study demonstrates that an appropriate combination of transverse roughness geometry and porous medium permeability can substantially enhance conjugate heat transfer performance in nanofluid-filled enclosures. These findings provide useful design guidelines for advanced thermal management systems, compact heat exchangers, and energy-efficient engineering applications
Rayleigh number Ra Porosity ϵ Nanoparticle volume fraction ϕ Solid conductivity ratio λk ​ Inclination angle δ
Mathematical Formulation on Fluid and Heat Transfer in Various Types of Channels
Fluid flow and heat transfer in channels constitute a fundamental area of research in mechanical, chemical, civil, and biomedical engineering. Mathematical modeling of transport phenomena in channels of different geometries—such as parallel plate channels, circular pipes, microchannels, and porous channels—provides critical insights into velocity distribution, temperature profiles, pressure drop, and thermal performance. This paper presents a comprehensive mathematical formulation governing fluid flow and heat transfer in various types of channels under laminar and turbulent conditions. The derivation is based on the conservation laws of mass, momentum, and energy, leading to the Navier–Stokes and energy equations. Dimensionless analysis using Reynolds, Prandtl, and Nusselt numbers is discussed to characterize flow regimes and thermal behavior. Special cases including fully developed flow, forced convection, and conjugate heat transfer are analyzed. The study highlights the impact of channel geometry, boundary conditions, and fluid properties on transport characteristics and provides a unified framework for further theoretical and computational investigations.
Fluid flow, Heat transfer, Channel flow, Navier–Stokes equations, Convection, Microchannels, Porous media, Mathematical modeling.
Exploring Analytical Conics through Traditional and Vedic Mathematical Frameworks
This study investigates analytical conic sections through a comparative examination of Traditional mathematical methods and Vedic mathematical techniques, emphasizing both procedural rigor and computational efficiency. The analysis reveals that traditional approaches provide systematic classification, algebraic precision, and strong geometric interpretation, yet often require lengthy multi-step derivations that may increase cognitive load for learners. In contrast, Vedic Mathematics offers efficient, pattern-based computational strategies that simplify algebraic manipulations and reduce procedural complexity. Techniques such as Urdhva–Tiryagbhyam and Paravartya Yojayet demonstrate particular effectiveness in factorization, solving simultaneous equations, and simplifying quadratic forms commonly encountered in conic problems.The findings indicate that Vedic methods function most effectively as complementary tools rather than replacements for traditional analytical frameworks. Integrating both approaches in instructional settings can enhance computational speed, minimize errors, and strengthen conceptual understanding while maintaining theoretical depth. This blended methodology also highlights the value of cross-cultural mathematical traditions in enriching modern analytical geometry. The study advocates for methodological pluralism in mathematics education, suggesting that the combined use of traditional rigor and innovative computational shortcuts can improve problem-solving skills, learner engagement, and pedagogical outcomes. Future research may explore applications in advanced geometrical contexts and empirical classroom implementations to further validate these findings.
Analytical geometry, conic sections, Vedic Mathematics, traditional methods, computational efficiency, mathematics education, algebraic manipulation, pedagogical innovation.
ANALYSIS OF TWO-PHASE FLOW DYNAMICS IN ARTERIES: INFLUENCE OF SODIUM CHLORIDE ON OXYGEN AND BLOOD TRANSPORT
Blood circulation represents a complex, multiphase system composed of plasma, red blood cells (RBCs), and dissolved gases such as oxygen (O₂) and carbon dioxide (CO₂). In a physiologically normal state, arterial blood maintains an oxygen saturation level between 95% and 100%, reflecting the proportion of hemoglobin molecules bound to oxygen and signifying adequate tissue oxygenation. The pulmonary artery, in contrast, transports oxygen-deficient blood from the right ventricle to the lungs, where gas exchange restores oxygen content. Oxygen saturation in this artery averages around 76% but rises close to 100% in the pulmonary veins once the blood has been re-oxygenated. The delivery of oxygen from arterial blood to peripheral tissues is influenced by hemodynamic factors and the electrolyte composition of plasma, particularly the concentration of sodium chloride (NaCl). This electrolyte governs osmotic balance, affects plasma viscosity, and alters gas diffusion rates, thereby modulating two-phase flow characteristics within the circulatory system. In this study, blood is conceptualized as a continuous phase with oxygen dispersed within it, and NaCl is incorporated into the model through its impact on viscosity and diffusion coefficients. Clinical data, including findings from the DECIDE-Salt trial, underline the broader physiological relevance of sodium regulation, linking excessive intake to elevated blood pressure and cardiovascular dysfunction. The integrated physiological–mathematical framework proposed here elucidates the dual influence of NaCl: enhancing micro-scale oxygen transport while concurrently modulating macro-scale cardiovascular dynamics and risk profiles.
Mathematical Models in Epidemiology: Bridging Theoretical Insights and Practical Applications
Mathematical models are very crucial in epidemiology, as they provide both theoretical insights and practical applications for public health interventions. This paper explores their transformative impact across five core areas: predicting disease outbreaks, evaluating public health interventions, integrating diverse data sources, addressing model uncertainty and parameter estimation, and analyzing the effects of model-driven policy decisions. This study uses a quantitative approach by analyzing independent variables such as model parameters and data inputs against the dependent variables that include prediction accuracy, intervention efficacy, and policy impact. Validations are carried out to check whether advanced models can indeed improve the accuracy of prediction, enhance assessment of interventions, and inform policies based on evidence. Despite considerable progress, real-time integration of data, quantification of uncertainty, and long-term reliability of models continue to pose significant challenges. Future research should focus on overcoming these limitations to fully realize the potential of mathematical models in advancing epidemiological research and public health outcomes.
Enhancing Hospital Operations and Patient Care: The Role of AI in Smart Healthcare Systems
This paper investigates the significance of differential equations in addressing complex engineering challenges, emphasizing their theoretical and practical roles in mechanical systems, fluid dynamics, electrical circuits, structural engineering, and control systems. The study utilizes a quantitative methodology to explore relationships between differential equation types and engineering outcomes such as model accuracy, system efficiency, and stability. Findings indicate that ordinary differential equations and partial differential equations are crucial in enhancing engineering solutions, testing hypotheses in different fields. These findings help bridge the gaps in research by pointing out the incorporation of differential equations into advanced models of engineering and future research directions to improve their applications in emerging technologies.
The Evolution of Calculus: Bridging Historical Foundations and Modern Computational Techniques
This paper traces the development of calculus-from its origins with pioneers Newton and Leibniz-to its use in neural networks and artificial intelligence. It looks into the original basic principles of calculus, their historical development, and transformation for contemporary computational demands. Amongst key results are comparative insights into the contributions of Newton and Leibniz and significant historical milestones regarding the integration of calculus into computational methods. The study also delves into the theoretical implications of calculus in machine learning and its profound influence on the advancement of AI. Employing a qualitative methodology, the research bridges historical context with modern applications, emphasizing the enduring relevance and adaptability of calculus in scientific and technological innovation.
The Role of Mathematical Sciences in Advancing Space Exploration: Challenges and Innovations
This study looks at the transformative potential of mathematical science in space travel by discussing, at a practical and theoretical level, some of the fields examined, from adaptive mathematical modeling of missions to optimizing resource use via novel techniques of resource allocation optimization. Astronomical data could also be examined more intensively by enhanced new methods, in addition to possibly establishing hybrid propulsion technologies. Utilizing a qualitative methodology, the research gathers insights from expert interviews and mission reports, applying thematic analysis to uncover patterns and themes. Findings highlight the critical contributions of mathematics in addressing challenges and driving innovations, emphasizing its role in improving mission outcomes, resource efficiency, and propulsion advancements. Despite its focus on specific missions, the study highlights the expansive potential of mathematics in space exploration and provides a foundation for future research to further explore its implications.
Innovations and Applications of Stochastic Processes in Financial Mathematics
This chapter explores the pivotal role of stochastic processes in financial mathematics, with an emphasis on their application in modeling market uncertainty and randomness. By investigating five key areas—option pricing, risk management, portfolio optimization, credit risk analysis, and the integration of stochastic processes with machine learning—it identifies how these processes innovate and transform financial theories and practices. Applying a quantitative methodology, the paper examines relationships between stochastic methods like differential equations and Monte Carlo approaches with dependent variables like accuracy in pricing, risk measures, and optimization results. The findings confirm that stochastic processes improve predictive precision, optimize financial strategies, and improve dynamic risk analysis. Findings also reveal the synergy between the potential of combining machine learning and stochastic methods to push financial modeling further. This chapter emphasizes stochastic processes as the very foundation driving innovation in financial mathematics while identifying gaps and future research opportunities for broader applicability.
Exploring the Interplay Between Topology and Graph Theory
This paper discusses the interrelation between topology and graph theory, with a focus on how these two disciplines blend together to solve complex problems in mathematics and practice. It discusses topological enhancement in graph-theoretical models, simplification of graph theory for topological issues, and interdisciplinary applications by computer science, biology, and physics. The discussion emphasizes important developments in theoretical mathematics and modeling. Findings indicate the establishment of uniform frameworks, such as topological graph theory, which combines discrete and continuous views. With considerable advancements, there is still no seamless integration, and hybrid approaches and cross-disciplinary collaboration are necessary. In this context, this paper stresses that the combination of topology and graph theory can potentially be revolutionary in the enhancement of theoretical understanding and in practice.
Understanding and applying chaos mathematics to complex systems
This study goes into the mathematical principles of chaos theory, focusing on its application to complex systems characterized by unpredictability and sensitivity to initial conditions. The literature review helped to raise key sub-research questions around nonlinearity, feedback loops, strange attractors, fractals, Lyapunov exponents, and the broader implications of chaos mathematics. The research applies a quantitative methodology, with statistical analysis, simulations, and mathematical modeling to study the role of chaos theory in real-world systems across fields like physics, biology, engineering, and economics. The results confirm several hypotheses: nonlinearity, feedback loops, and strange attractors are significant, but the practical utility of fractals and Lyapunov exponents in system analysis is also underlined. The findings confirm that chaos mathematics provides transformational insights-from weather forecasting to cryptography-that have a potential application.
Exploring the Integral Role of Probability in Risk Assessment and Prediction
This study investigates the central role of probability in assessing and predicting risk and provides evidence to its applicability in insurance, engineering, meteorology, economics, and complex decision-making. By exploring probabilistic methods and their impact on the accuracy of predictions, the study shows how an integration of advanced tools like machine learning, big data analytics, and Bayesian models improves precision across all domains. Using data from 2000–2023, regression analyses validate five hypotheses, confirming significant improvements in risk estimation and prediction reliability. The findings underline the theoretical and practical importance of probability, bridging critical gaps in long-term model efficacy, risk mitigation strategies, and advanced tool integration.
The Symmetry in Mathematics: Exploring Patterns in Nature and Mathematics
The research explores the intricate interplay between symmetry in natural phenomena and mathematical theories, putting emphasis on its aesthetic, biological, and technological importance. Qualitative methodologies have been used to investigate five sub-research areas, namely types of natural symmetries, mathematical principles underlying symmetry, biological functions of symmetry, aesthetic appeal, and technological applications. The results indicated that natural symmetries are more complex than before, mathematical models of this are changing to take on complex natural patterns, and symmetry is critical to both biological functionality and aesthetic appreciation. Moreover, the application of symmetry in novel technologies has shown its tremendous scope. These findings and facts have highlighted the multi-faceted importance of symmetry and its wide reach beyond disciplines, thus disqualifying earlier simplistic classification of this notion.
Gravitational Wave Probes of Modified Gravity: A new Tool for testing Cosmological Models
Gravitational-wave (GW) observation has opened up a new array of information on the universe by adding information that has been obtained from electromagnetic and particle astronomy observations. Not only can physicists develop further confidence in Einstein’s General Relativity(GR) through GW’s, but GW’s can also be used to conduct ‘tests’ on alternative theories of gravity that have previously been unimaginable or impossible to test. Many of these theories of modified gravity (MG), which have been suggested to explain the apparent acceleration of the cosmos and the nature of dark matter, or provide an explanation for deficiencies in the Standard Model, predict that GW’s would have fundamental differences in the way they are produced, and the way they propagate through space-time, as well as differences in their polarization. In this article, we discuss how current and future GW data can provide a means of testing such MG theories. We provide a brief overview of the theoretical constraints and the most recent published results from the LIGO-Virgo-KAGRA collaboration, as well as discussing an upcoming way to evaluate cosmological models via GW propagation effects and multi-messenger detection, along with likely deviations in waveform characteristics. Our analysis indicates that by using future detectors, such as LISA and the Einstein Telescope, MG parameter constraints will be greatly reduced, thereby providing a means to unequivocally differentiate GR from alternative gravitational frameworks.
Scalar–Tensor Theories and their Implications for Early Universe Cosmology
Scalar–tensor theories of gravitation represent one of the most intuitive and mathematically coherent extensions of Einstein’s General Relativity (GR). These theories augment the traditional tensor framework of gravity by integrating one or more scalar fields, thereby introducing additional complexity to the dynamics of space-time geometry and the interactions of matter. Their significance in cosmology is considerable, particularly in the context of the early universe. By providing additional degrees of freedom, scalar– tensor models help to resolve some of the key puzzles of standard cosmology, including the initial singularity, horizon and flatness problems, and the mechanism driving the universe’s rapid inflationary expansion. The scalar field, which may be coupled either minimally or non-minimally to curvature, is pivotal in inflationary models by facilitating a nearly de Sitter phase that mitigates inhomogeneities and initiates the large-scale structure of the Universe. Beyond the inflationary period, scalar–tensor dynamics significantly impact the reheating phase, dictate the evolution of primordial perturbations, and leave distinct signatures in the Cosmic Microwave Background (CMB) and gravitational wave background. In addition, these models propose a theoretical framework to integrate early universe physics with late-time acceleration phenomena associated with dark energy, suggesting that the same scalar degree of freedom could be responsible for both cosmic inflation and the current expansion of the universe. This paper provides a comprehensive examination of scalar–tensor theories in the context of early universe cosmology. We begin with a detailed presentation of their mathematical framework, discuss their implications for inflationary dynamics, reheating, and perturbation growth, and assess the ways in which they modify key observational predictions. Furthermore, we highlight the theoretical challenges, such as frame dependence, the necessity for precise adjustments, and strict astrophysical constraints on scalar couplings. Finally, we examine the potential for future observational validation, emphasizing how upcoming cosmological surveys and gravitational wave experiments may either corroborate or constrain the impact of scalar–tensor gravity on the evolution of the universe.
Approach to Solving Stochastic Differential Equations with Jumps Using Adaptive Time-Stepping and High-Order Milstein Schemes
This paper presents a novel and efficient numerical method for solving Stochastic Differential Equations with Jumps (SDEJs). We introduce an adaptive time-stepping scheme coupled with a high-order Milstein approximation to enhance the accuracy and stability of the solution. The adaptive time-stepping is designed to dynamically adjust the step size based on the local behavior of the solution, thereby improving computational efficiency. The high-order Milstein scheme is tailored to handle the jump component effectively, particularly when the jump sizes are significant. We provide a rigorous convergence analysis of the proposed method and demonstrate its superior performance through numerical experiments. The results show that the adaptive Milstein scheme offers a significant improvement in accuracy and efficiency compared to traditional fixed-step methods, especially for SDEJs with large jump intensities.
Stochastic Differential Equations with Jumps (SDEJs)
A Novel Hybrid Metaheuristic Approach for Solving High-Dimensional Global Optimization Problems with Enhanced Exploration-Exploitation Balance
This paper presents a novel hybrid metaheuristic algorithm designed to address the challenges posed by high-dimensional global optimization problems. The algorithm synergistically combines the strengths of Particle Swarm Optimization (PSO) and Differential Evolution (DE) with an adaptive control mechanism to dynamically balance exploration and exploitation. The hybrid approach leverages PSO's efficient global search capability and DE's effective local refinement to achieve enhanced performance. The adaptive control mechanism adjusts the contributions of PSO and DE based on the search progress, promoting exploration in the early stages and intensifying exploitation as the search converges. The performance of the proposed algorithm is evaluated on a suite of benchmark functions, including unimodal, multimodal, and composite functions, and compared against established metaheuristic algorithms. The results demonstrate the superior performance of the hybrid algorithm in terms of solution accuracy, convergence rate, and robustness, particularly in high-dimensional spaces.
A Novel Hybrid Approach Integrating Fractional Calculus and Deep Learning for Enhanced Time Series Forecasting of Chaotic Systems
This paper introduces a novel hybrid approach for time series forecasting of chaotic systems, integrating the strengths of fractional calculus and deep learning. Chaotic systems, characterized by their sensitive dependence on initial conditions, pose significant challenges for accurate prediction. While deep learning models, particularly Long Short-Term Memory (LSTM) networks, have shown promise in capturing complex temporal dependencies, they often struggle with long-range dependencies and noise inherent in chaotic data. We propose a hybrid model that leverages fractional derivatives to enhance the representation of past states, thereby improving the LSTM network's ability to learn and forecast chaotic time series. The fractional derivative captures non-local dependencies more effectively than traditional integer-order derivatives, providing richer information for the deep learning component. We evaluate the performance of our proposed model on benchmark chaotic systems, including the Lorenz attractor and the Rossler system. The results demonstrate that our hybrid approach significantly outperforms traditional LSTM networks and other established forecasting methods in terms of prediction accuracy, especially over longer forecasting horizons. This work provides a valuable contribution to the field of time series forecasting for chaotic systems, offering a powerful tool for modeling and predicting complex dynamical behaviors.
A Novel Hybrid Approach Combining Fractal Analysis and Machine Learning for Enhanced Prediction of Chaotic Time Series
Predicting chaotic time series remains a significant challenge due to their inherent sensitivity to initial conditions and complex nonlinear dynamics. This paper introduces a novel hybrid approach that combines fractal analysis techniques with machine learning models to improve prediction accuracy. Specifically, we leverage fractal dimension estimation, recurrence plot analysis, and the Hurst exponent to extract key features from chaotic time series. These features are then used as inputs to a Support Vector Regression (SVR) model. The efficacy of this hybrid method is demonstrated through extensive experimentation on benchmark chaotic time series datasets, including the Lorenz attractor, Rossler attractor, and Mackey-Glass equation. Results indicate that the proposed approach significantly outperforms traditional time series prediction methods, offering a robust and accurate framework for forecasting chaotic dynamics. This hybrid strategy effectively captures both the local and global characteristics of chaotic systems, leading to enhanced predictive performance.
A Novel Hybrid Approach Combining Fractional Calculus and Metaheuristic Optimization for Solving High-Dimensional Non-Linear Partial Differential Equations
This paper presents a novel hybrid approach for solving high-dimensional non-linear partial differential equations (PDEs) by integrating fractional calculus concepts with metaheuristic optimization algorithms. Specifically, we employ the Caputo fractional derivative to model the PDE and then utilize a modified Particle Swarm Optimization (PSO) algorithm to minimize the error functional associated with the fractional PDE. The proposed method addresses the challenges posed by high dimensionality and non-linearity, which often render traditional numerical techniques computationally infeasible or inaccurate. We demonstrate the efficacy and accuracy of our approach through several benchmark problems, comparing our results with those obtained by existing methods. The convergence analysis and computational efficiency of the hybrid algorithm are also investigated. The results demonstrate that the proposed method offers a promising alternative for solving complex fractional PDEs in various scientific and engineering applications.
Ellipticity of Operators Under Shear Transformations
This work investigates nonlocal boundary-value problems in which the primary operator and the operators within the boundary conditions encompass differential operators and twisting transformations. These problems extend classical boundary-value problem frameworks by incorporating nonlocal effects and complex interactions introduced through twisting operators. A novel definition of trajectory symbols is presented, tailored to address the unique characteristics of this class of problems.We demonstrate that the elliptic nature of these problems ensures the generation of Fredholm operators in the appropriate Sobolev spaces. This result establishes well-posedness, confirming that the problems admit solutions that depend continuously on the given data. Furthermore, we provide a comprehensive ellipticity condition specific to these nonlocal problems, serving as a key criterion for their solvability.The findings expand the theory of elliptic boundary-value problems by including settings with nonlocal and shear-like operators, offering insights into their analytical structure. This work contributes to the broader understanding of boundary-value problems with complex operator interactions and provides a foundation for future investigations into more generalized operator frameworks and applications.
Matter Dynamics in Helical Magnetic Fields: A Mathematical Model with Infinite Boundaries
This paper develops a mathematical model of plasma transfer in an open magnetic trap, incorporating the boundary condition of zero plasma concentration at infinity. Experimental data from the SMOLA trap at the Budker Institute of Nuclear Physics SB RAS serve as validation for the model. Plasma confinement is achieved by transferring momentum from a helical-symmetric magnetic field to a rotating plasma. The model is formulated using a stationary plasma transfer equation in an axially symmetric configuration, incorporating second-order spatial derivatives. An optimal numerical template for approximating mixed derivatives is determined through a benchmark test problem. The numerical implementation of the model is evaluated by comparing the establishment method and the successive over-relaxation method. The findings demonstrate the model’s capability to accurately describe plasma dynamics in helical magnetic fields and its effectiveness in optimizing plasma confinement.
Topological Insights into Prime and Minimal Prime Filters on Paradistributive Latticoids
In this paper, we investigate prime filters and minimal prime filters on a paradistributive latticoid (PDL) and establish significant theoretical results. We demonstrate that the annihilator filter S∙S^\bullet is equal to the intersection of all prime filters not containing SS, providing a novel perspective on filter structures in PDLs. Furthermore, we explore minimal prime filters, analyzing their unique properties and relationships within the PDL framework. Our study also presents equivalent conditions for a PDL to be relatively complemented, offering insights into its algebraic structure. Additionally, we derive and discuss the topological properties of the spaces of prime filters and minimal prime filters, emphasizing their interactions and contributions to the broader understanding of PDLs. These results expand the foundational knowledge of PDLs and provide new tools for analyzing their algebraic and topological characteristics.
Geometric Characterizations of Euclidean Spheres Using the Tangential Component of the Position Vector
In this study, we explore the geometric properties of spheres on a compact hypersurface in Euclidean space Rn+1\mathbb{R}^{n+1} using the support function θ and the tangential component ψT\psi_T of the position vector field ψ\psi. The first characterization extends existing results by removing constraints on the tangential component ψT\psi_T, and applies alternative proof techniques to obtain new insights. In the second characterization, we focus on the specific role of the support function θ in defining these geometric properties, providing a more comprehensive understanding of the structure of spheres within the hypersurface. This work contributes to the broader field of differential geometry by offering novel approaches to the study of hypersurfaces and their associated curvature characteristics.
An Investigation into the Nonexistence of Nontrivial Weak Solutions of Nonlinear Inequalities with Gradient Nonlinearities
In this article, we extend and refine the work of Mitidieri and Pohozhaev concerning the nonexistence of nontrivial weak solutions to nonlinear inequalities and systems. Specifically, we focus on problems involving integer powers of the Laplace operator and nonlinear terms of the form a(x)∣∇(Δmu)∣q+b(x)∣∇u∣sa(x) |\nabla(\Delta^m u)|^q + b(x) |\nabla u|^s. By leveraging the nonlinear capacity method and carefully selecting suitable test functions, we derive optimal a priori estimates. These estimates allow us to prove, via contradiction, the nonexistence of nontrivial weak solutions to the given nonlinear inequalities and systems. Our results provide new insights into the conditions under which solutions fail to exist, contributing to the broader understanding of nonlinear partial differential equations with gradient nonlinearity.