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Adaptive Time Stepping Numerical Schemes for Stochastic Differential Equations Rishav Jha; Kameshwar Sahani; Suresh Kumar Sahani; Ravi Kumar Raj; Dilip Kumar Sah
Asian Journal of Science, Technology, Engineering, and Art Vol 4 No 3 (2026): Asian Journal of Science, Technology, Engineering, and Art
Publisher : Darul Yasin Al Sys

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.58578/ajstea.v4i3.10239

This study presents a comprehensive examination of adaptive time-stepping numerical schemes for solving stochastic differential equations (SDEs), with particular attention to methods that automatically adjust step sizes based on local error estimates. The study aims to investigate the theoretical foundations, implementation strategies, convergence properties, and practical applications of adaptive numerical methods for SDEs. The Euler–Maruyama and Milstein schemes were extended through adaptive step-size control mechanisms, and their convergence behavior was analyzed through extensive numerical experiments implemented in Python. The study also provides detailed code examples, accessible explanations, and visualizations, including convergence plots, error analysis, and performance comparisons, to support practical understanding and implementation. The findings indicate that adaptive schemes substantially improve computational efficiency while maintaining required levels of accuracy. Specifically, the results show that adaptive methods can reduce computational costs by up to 60% compared with fixed-step methods for problems involving varying stiffness. The study concludes that adaptive time-stepping offers a robust and efficient strategy for numerical SDE simulation, particularly in computational settings where accuracy and efficiency must be balanced. Its contribution lies in integrating theoretical analysis, implementation guidance, and empirical performance evaluation to support researchers and practitioners in applying adaptive numerical schemes to stochastic differential equations.

Data-Driven Identification of Stochastic Dynamical Systems Rishav Jha; Kameshwar Sahani; Suresh Kumar Sahani; Ravi Kumar Raj; Dilip Kumar Sah
African Multidisciplinary Journal of Sciences and Artificial Intelligence Vol 3 No 2 (2026): African Multidisciplinary Journal of Sciences and Artificial Intelligence
Publisher : Darul Yasin Al Sys

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.58578/amjsai.v3i2.10238

Identifying stochastic dynamical systems from observational data remains a major challenge in applied mathematics and engineering, particularly when complex systems are influenced by random perturbations and incomplete empirical information. This comprehensive review aims to examine state-of-the-art data-driven methods for discovering governing equations, estimating parameters, and predicting the behavior of stochastic dynamical systems. The review systematically analyzes key methodological approaches, including Sparse Identification of Nonlinear Dynamics (SINDy), Dynamic Mode Decomposition (DMD) and its extensions, Koopman operator theory, neural ordinary differential equations, and Bayesian inference. Each approach is evaluated in terms of its theoretical foundations, computational requirements, robustness to noise, and applicability to different classes of stochastic systems. Drawing on numerical experiments and real-world case studies, the findings show that no single method consistently outperforms others across all scenarios. Instead, hybrid approaches that integrate physics-informed constraints with machine learning demonstrate the strongest potential for advancing data-driven system identification. The review concludes that future research should address real-time identification, uncertainty quantification, and the integration of multi-fidelity data sources to improve the reliability and scalability of stochastic system modeling. This work contributes a comprehensive framework for guiding researchers and practitioners in selecting and implementing appropriate identification methods for stochastic dynamical systems.

Data-Driven Identification of Stochastic Dynamical Systems Rishav Jha; Kameshwar Sahani; Suresh Kumar Sahani; Ravi Kumar Raj; Dilip Kumar Sah
African Multidisciplinary Journal of Sciences and Artificial Intelligence Vol 3 No 2 (2026): African Multidisciplinary Journal of Sciences and Artificial Intelligence
Publisher : Darul Yasin Al Sys

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.58578/amjsai.v3i2.10238

Identifying stochastic dynamical systems from observational data remains a major challenge in applied mathematics and engineering, particularly when complex systems are influenced by random perturbations and incomplete empirical information. This comprehensive review aims to examine state-of-the-art data-driven methods for discovering governing equations, estimating parameters, and predicting the behavior of stochastic dynamical systems. The review systematically analyzes key methodological approaches, including Sparse Identification of Nonlinear Dynamics (SINDy), Dynamic Mode Decomposition (DMD) and its extensions, Koopman operator theory, neural ordinary differential equations, and Bayesian inference. Each approach is evaluated in terms of its theoretical foundations, computational requirements, robustness to noise, and applicability to different classes of stochastic systems. Drawing on numerical experiments and real-world case studies, the findings show that no single method consistently outperforms others across all scenarios. Instead, hybrid approaches that integrate physics-informed constraints with machine learning demonstrate the strongest potential for advancing data-driven system identification. The review concludes that future research should address real-time identification, uncertainty quantification, and the integration of multi-fidelity data sources to improve the reliability and scalability of stochastic system modeling. This work contributes a comprehensive framework for guiding researchers and practitioners in selecting and implementing appropriate identification methods for stochastic dynamical systems.

Deep Learning-Based Approximation of Solutions to Stochastic Differential Equations Rishav Jha; Kameshwar Sahani; Suresh Kumar Sahani; Ravi Kumar Raj; Dilip Kumar Sah
International Journal of Education, Management, and Technology Vol 4 No 2 (2026): International Journal of Education, Management, and Technology
Publisher : Darul Yasin Al Sys

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.58578/ijemt.v4i2.10241

Stochastic differential equations (SDEs) are essential mathematical tools for modeling systems subject to random influences across finance, physics, biology, and engineering. However, traditional numerical methods, including the Euler–Maruyama and Milstein schemes, face substantial limitations in high-dimensional settings and often require extensive Monte Carlo simulations to obtain accurate statistical estimates. This study aims to develop and evaluate a deep learning framework for approximating SDE solutions using Physics-Informed Neural Networks (PINNs) and Deep Backward Stochastic Differential Equation methods. The proposed methodology leverages automatic differentiation to enforce the underlying stochastic dynamics through a composite loss function incorporating PDE residuals, boundary conditions, and initial conditions. The framework was assessed through benchmark problems, including geometric Brownian motion, Ornstein–Uhlenbeck processes, and the Black–Scholes equation. The findings indicate that deep learning approaches achieve superior accuracy compared with traditional numerical schemes while offering substantial computational advantages, particularly for high-dimensional problems. Experimental results show that the proposed approach achieves relative errors below 1% and provides speedup factors exceeding 100 times for 50-dimensional problems compared with conventional Monte Carlo methods. The study concludes that PINNs and Deep BSDE methods offer a promising computational paradigm for solving high-dimensional stochastic differential equations efficiently and accurately. This work contributes to scientific machine learning and numerical SDE research by demonstrating the potential of deep learning-based solvers to address dimensionality-related limitations in conventional stochastic simulation methods.

Geometric Foundations of Engineering Design: The Role of Conic Sections Enhanced by Artificial Intelligence Rajnandani Das; Neha Shah; Suresh Kumar Sahani
Asian Journal of Science, Technology, Engineering, and Art Vol 4 No 1 (2026): Asian Journal of Science, Technology, Engineering, and Art
Publisher : Darul Yasin Al Sys

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.58578/ajstea.v4i1.8700

Many branches of engineering rely on four fundamental geometric shapes: circle, ellipse, parabola, and hyperbola, whose intrinsic properties enable engineers to develop more accurate mathematical models, optimize trajectories, and enhance structural integrity in complex design contexts. This study examines how these classical conic sections are applied in real-world engineering problems and explores the utilization of geometric principles in robotics, signal processing, and automated systems to support efficient problem-solving. By relating the properties of conic sections to engineering requirements in areas such as bridge design, trajectory optimization, and structural analysis, the study elucidates how these forms underpin both analytical modelling and practical implementation in contemporary engineering practice. The analysis shows that the relevance of conic sections to practical engineering applications is clearly demonstrated across multiple domains, highlighting their role in improving modelling accuracy, guiding system optimization, and informing robust design strategies. The study concludes that classical geometry, particularly the theory of conic sections, continues to play a vital role in shaping modern engineering practices and carries important implications for advancing engineering education, promoting interdisciplinary integration, and sustaining innovation in technology and infrastructure development.

Deep Learning - Based Shape Recognition and Classifications of Conic Geometries in Engineering Drawing Rajnandani Das; Neha Shah; Dilip Kumar Sah; Kameshwar Sahani; Suresh Kumar Sahani
Asian Journal of Science, Technology, Engineering, and Art Vol 4 No 2 (2026): Asian Journal of Science, Technology, Engineering, and Art
Publisher : Darul Yasin Al Sys

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.58578/ajstea.v4i2.9335

Engineering drawings frequently contain conic geometries such as circles, ellipses, parabolas, and hyperbolas, which are fundamental to mechanical design and industrial applications. Accurate identification and classification of these shapes are therefore essential for computer-aided design (CAD) systems, automated inspection, and intelligent design analysis. However, conventional geometry-based or rule-based approaches often perform poorly when drawings are noisy, complex, or partially incomplete. This study proposes a deep learning-based approach using convolutional neural networks (CNNs) to automatically extract features and classify conic shapes in engineering drawings. By learning discriminative visual representations directly from input data, the proposed method enhances classification accuracy, improves robustness, and reduces the need for manual intervention. The study concludes that CNN-based conic shape recognition offers a reliable and efficient solution for engineering and industrial contexts, with practical implications for improving automation and intelligent analysis in design-related applications.

Utilizing Permutation and Combination Techniques in Business Decision-Making Processes Bardan Sah; Ritika Jayswal; Satyam Thakur; Neha Shah; Dilip Kumar Sah; Suresh Kumar Sahani
Mikailalsys Journal of Mathematics and Statistics Vol 4 No 2 (2026): Mikailalsys Journal of Mathematics and Statistics
Publisher : Darul Yasin Al Sys

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.58578/mjms.v4i2.9336

Although permutations and combinations are often regarded as purely theoretical mathematical topics, they play a significant role in practical decision-making and contemporary business operations. This study examines the application of permutations and combinations in everyday decision-making and real business contexts, particularly in quality control, marketing strategy, resource planning, and inventory management. Using real-world examples and case studies, the article demonstrates how organizations employ these combinatorial concepts to improve productivity, reduce costs, optimize available resources, and strengthen competitive advantage in increasingly complex market environments. The findings indicate that a sound understanding of permutations and combinations enhances managerial and executive decision-making, especially when evaluating numerous alternatives, assessing the likelihood of possible outcomes, selecting appropriate combinations of people or products, and determining optimal configurations. The study concludes that permutations and combinations are not merely academic concepts but practical analytical tools that support more effective and strategic business decisions. This study contributes to a broader understanding of how foundational mathematical reasoning can be applied to improve organizational efficiency and decision quality in business practice.

Practical Use of Derivatives in Different Engineering Fields Praveen Shah; Suresh Kumar Sahani
Mikailalsys Journal of Advanced Engineering International Vol 3 No 1 (2026): Mikailalsys Journal of Advanced Engineering International
Publisher : Darul Yasin Al Sys

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.58578/mjaei.v3i1.8274

This study examines the practical applications of derivatives as a cornerstone of engineering mathematics, enabling precise quantification of rates of change, optimization of processes, and accurate prediction of system behavior. It aims to demonstrate how derivatives are operationalized across mechanical, electrical, civil, chemical, aerospace, and computer engineering through motion analysis, structural integrity assessment, electrical circuit dynamics, chemical reaction rate calculations, flight stability analysis, and computational optimization. Employing case studies, mathematical models, and real-world examples, the paper systematizes the role of derivatives in formulating and solving engineering problems, supported by illustrative tables, formulas, and graphs that clarify key computational steps and outcomes. The findings highlight that derivatives provide a unifying analytical framework for modeling dynamic phenomena, improving design reliability, and enhancing control and optimization strategies across diverse engineering domains. The study concludes that a rigorous understanding and applied use of derivatives are essential for effective engineering analysis and decision-making, with implications for strengthening curricula in engineering mathematics and promoting derivative-based approaches in professional engineering practice.

Time as Dimension or Illusion? A Critical Analysis within the Framework of Relativity Praveen Sah; Neha Shah; Dilip Kumar Sah; Suresh Kumar Sahani
Mikailalsys Journal of Advanced Engineering International Vol 3 No 1 (2026): Mikailalsys Journal of Advanced Engineering International
Publisher : Darul Yasin Al Sys

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.58578/mjaei.v3i1.9337

The nature of time remains a central problem in both physics and philosophy, particularly in light of the tension between classical and relativistic conceptions of temporality. This paper examines the question of whether time is an illusion within the framework of relativity. Whereas classical physics treats time as absolute, universal, and uniformly flowing, Einstein’s theory of relativity demonstrates that temporal intervals vary according to relative motion and gravitational fields. Building on this framework, the paper argues that time is operationally real insofar as it can be measured and modeled physically, yet the notion of a universally shared and continuously flowing present has no firm basis in modern physics. The analysis further suggests that the relativistic view of spacetime supports the coexistence of past, present, and future within a unified four-dimensional structure. It also considers whether the human experience of temporal passage arises from fundamental physical laws or from cognitive and thermodynamic asymmetries. The paper concludes that time itself is not an illusion; rather, what is misleading is the classical intuition that time flows identically for all observers. This study contributes to ongoing interdisciplinary debates by clarifying how relativity reshapes the philosophical interpretation of temporal reality.

Model-Free Reinforcement Learning for Parabolic Trajectory Optimization in Robotic Arms Aadarsh Karn; Neha Shah; Dilip Kumar Sah; Suresh Kumar Sahani
African Multidisciplinary Journal of Sciences and Artificial Intelligence Vol 3 No 1 (2026): African Multidisciplinary Journal of Sciences and Artificial Intelligence
Publisher : Darul Yasin Al Sys

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.58578/amjsai.v3i1.9338

Robotic arms are widely employed in applications that require smooth motion and energy-efficient operation, particularly in tasks such as object throwing and liquid dispensing, where movement often follows a curved path toward a target point. However, conventional trajectory planning methods that rely on predefined mathematical equations may not accurately represent real-world robotic systems due to uncertainties and payload variations. This study aims to optimize the trajectory of a robotic arm moving along a parabolic path using reinforcement learning and to evaluate whether this approach can successfully learn improved trajectory patterns during motion. The research integrates initial classical physics principles for curved motion with a reinforcement learning framework to enhance trajectory following toward a desired point. The findings indicate that reinforcement learning can effectively learn optimized trajectory paths and improve the motion performance of the robotic arm. The study concludes that reinforcement learning offers a promising approach for achieving smoother robotic motion with satisfactory energy efficiency under dynamic conditions. This work contributes to the advancement of intelligent motion planning by demonstrating the potential of reinforcement learning to improve trajectory optimization in robotic systems operating under practical uncertainties.

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