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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