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Research Article
A Block-based Linear Multistep Formula for Directly Solving Nonlinear Fourth-order Initial Value Problems of ODEs
Issue:
Volume 13, Issue 2, April 2025
Pages:
103-116
Received:
13 January 2025
Accepted:
27 January 2025
Published:
27 February 2025
Abstract: This paper suggested a block-based linear multistep formula for directly solving nonlinear fourth-order initial value problems of ordinary differential equations (ODEs). The method was achieved by applying collocation and interpolation techniques to a first-kind Chebyshev polynomial. A continuous scheme was constructed through this procedure from where the proposed discrete formula was extracted. The extracted discrete formula was then implemented in block mode using the block matrix formulation and written explicitly as block equations. The proposed method is zero-stable, consistent, convergent, and p-stable, as demonstrated by the analysis of the basic properties of the derived scheme, with theoretical order eight. Six numerical examples were solved with the derived method to test its accuracy and effectiveness, all showing minimal error. A comparison with existing methods in the cited literature revealed that the proposed method offers good performance with minor errors.
Abstract: This paper suggested a block-based linear multistep formula for directly solving nonlinear fourth-order initial value problems of ordinary differential equations (ODEs). The method was achieved by applying collocation and interpolation techniques to a first-kind Chebyshev polynomial. A continuous scheme was constructed through this procedure from w...
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Research Article
Navigating the Fourth Dimension: Relativity and Perception Through a 3D Lens
Charde’Lyce Edwards*
Issue:
Volume 13, Issue 2, April 2025
Pages:
117-124
Received:
22 January 2025
Accepted:
5 February 2025
Published:
11 March 2025
DOI:
10.11648/j.ajam.20251302.12
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Abstract: Perception is inherently constrained by dimensionality, limiting a three-dimensional (3D) observer’s ability to interpret four-dimensional (4D) structures. While human experience is confined to three spatial dimensions with time perceived as a linear progression, relativistic principles—such as Lorentz transformations and spacetime curvature—suggest more complex interactions in higher dimensions. By integrating mathematical modeling with relativistic physics, this study examines how 3D observers might infer 4D structures and the challenges that arise when engaging with projections of higher-dimensional phenomena. Utilizing thought experiments, the consideration of spatial distortions, cross-sectional representations, and dimensional and how these limit direct comprehension of 4D objects. Additionally, relativistic effects, such as time dilation, frame-dependent simultaneity, and non-Euclidean spatial transformations, may influence temporal perception in a 4D framework, challenging conventional notions of sequential time. The inability to directly visualize or intuitively grasp higher-dimensional structures underscores the fundamental cognitive and perceptual barriers inherent in dimensional inference. Beyond theoretical physics, these insights extend to computational modeling, virtual reality, and quantum information science. Understanding how lower-dimensional observers infer higher-dimensional structures could inform new approaches to spatial computing, immersive simulations, and advanced visualization techniques. By bridging physics, mathematics, and perception, this research deepens the exploration of multidimensional reality, offering perspectives that may influence future developments in both scientific thought and technological innovation.
Abstract: Perception is inherently constrained by dimensionality, limiting a three-dimensional (3D) observer’s ability to interpret four-dimensional (4D) structures. While human experience is confined to three spatial dimensions with time perceived as a linear progression, relativistic principles—such as Lorentz transformations and spacetime curvature—sugges...
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Research Article
New Symmetry Index Based on Gini Mean Difference
Eman Mohamed Hanafy*,
Hend Abdulghaffar Auda,
Ibrahim Hassan Ibrahim
Issue:
Volume 13, Issue 2, April 2025
Pages:
125-142
Received:
13 February 2025
Accepted:
25 February 2025
Published:
18 March 2025
DOI:
10.11648/j.ajam.20251302.13
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Abstract: The Gini index is a widely used tool for measuring inequality, but it has several limitations that can lead to misinterpretation or incorrect conclusions, as highlighted in various studies. A significant drawback of the Gini index is that it fails to account for crucial aspects of inequality, such as the heterogeneity within a population, and the asymmetry of the data, meaning how skewed or unbalanced the distribution may be. In response to these shortcomings, a new index has been developed that more accurately captures both inequality and the symmetry of data. This new index builds on Auda's symmetry test and leverages a mathematical relationship between the Gini mean difference and the Gini index, providing a more refined measure. Through a Monte Carlo simulation, the new index demonstrated its superiority over existing ones, as it effectively reveals the distribution of asymmetrical data (whether positively or negatively skewed). Unlike the Gini index, this new index can differentiate between datasets with identical Gini values but different levels of symmetry. Additionally, it is more versatile, able to be applied to datasets of any size, including those that contain negative values. The index’s effectiveness is demonstrated with examples, including a scenario where two populations have the same total income and an educational study using data from Helwan University’s Faculty of Social Work.
Abstract: The Gini index is a widely used tool for measuring inequality, but it has several limitations that can lead to misinterpretation or incorrect conclusions, as highlighted in various studies. A significant drawback of the Gini index is that it fails to account for crucial aspects of inequality, such as the heterogeneity within a population, and the a...
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Research Article
Natural Convective Flow and Heat Transfer in a U-Shaped Device with a Solid Strip Using Al2O3-Water Nanofluid
Main Uddin Ahammad*,
Shohag Hossain Reyad
Issue:
Volume 13, Issue 2, April 2025
Pages:
143-152
Received:
17 February 2025
Accepted:
28 February 2025
Published:
18 March 2025
DOI:
10.11648/j.ajam.20251302.14
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Abstract: This research explores the numerical investigation of natural convective heat transfer in a U-shaped device having an internal thermally isolated solid strip through the use of nanofluid. Such configuration along with specified boundary conditions is very demanding for obtaining most favourable cooling efficiency. The nanoparticle alumina (Al2O3) is considered here to form single-phase nanofluid mixing with pure water corresponds to Pr = 7.0. Galerkin weighted residual based finite element method along with sophisticated software has been adopted for solving the non-dimensional partial differential equations (continuity of mass, momentum and energy) that govern the present problem. The effects of natural convection parameter Rayleigh number varied as 104 ≤ Ra ≤107 and geometric parameter volume fraction of nanoparticle in the range 0.01 ≤ φ ≤ 0.1 on the flow and thermal field as well as heat transfer rate have been analyzed and expressed by streamlines, isotherms and average Nusselt number. Moreover, for better understanding of flow visualization and temperature behaviour velocity profile, temperature profile and temperature gradient magnitude profile are also exposed. Major outcomes of the current work are displayed in both of the tabular and graphical form. The results indicate that the average Nusselt number which is the representative of heat transfer performance rises as both of Rayleigh number and the nanoparticle volume fraction increases which establish the significance of pertinent parameters in respective field.
Abstract: This research explores the numerical investigation of natural convective heat transfer in a U-shaped device having an internal thermally isolated solid strip through the use of nanofluid. Such configuration along with specified boundary conditions is very demanding for obtaining most favourable cooling efficiency. The nanoparticle alumina (Al2O3) i...
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Research Article
Optimal Control of a Nonlinear System with White Noise
Georges Kologo*,
Cédric Kpèbbèwèrè Some,
Somdouda Sawadogo
Issue:
Volume 13, Issue 2, April 2025
Pages:
153-164
Received:
19 February 2025
Accepted:
10 March 2025
Published:
26 March 2025
DOI:
10.11648/j.ajam.20251302.15
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Abstract: In this paper, we propose a control problem for nonlinear stochastic differential equations with noise. The system proposed for the control problem is a nonlinear system perturbed by standard Brownian motion. We write this problem in a coupled form where the control u(t) has a value in a convex space. Here, we propose sufficient minimum conditions on the Hamiltonian function to characterize the mean value of the cost function associated with the optimal control problem. The convexity of the Hamiltonian function is a sufficient condition for the existence of the optimal value of the control function. Under certain regularity assumptions, on the control system functional and on the non-differentiability criterion of Brownian motion, the existence and uniqueness results are established by the Cauchy-Lipschitz criteria. We also analyze the mathematical expectation stability of the system to check whether it will converge to an equilibrium point or not. For the study of this stability, the emphasis has been placed on root-mean-square stability. To highlight the results of our work, we apply this control problem to a SIRS-type epidemiological system for the coronavirus epidemic. To study the stability of this epidemiological system, we construct a Lyaunov function associated with the system and then use the results of Lyapunov's theorem to show the convergence of the system to a stable state.
Abstract: In this paper, we propose a control problem for nonlinear stochastic differential equations with noise. The system proposed for the control problem is a nonlinear system perturbed by standard Brownian motion. We write this problem in a coupled form where the control u(t) has a value in a convex space. Here, we propose sufficient minimum conditions ...
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