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Application of the Reduced Differential Transform Method to Solve the Navier-Stokes Equations
Yanick Alain Servais Wellot
Issue:
Volume 11, Issue 6, December 2022
Pages:
96-101
Received:
18 October 2022
Accepted:
12 November 2022
Published:
22 November 2022
DOI:
10.11648/j.pamj.20221106.11
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Abstract: In fluid mechanics, the Navier-Stokes equations are non-linear partial differential equations that describe the motion of Newtonian fluids. A fluid can be a liquid or a gas. Therefore, the Navier-Stokes equation concerns many phenomena that surround us. The analytical resolution, the search for exact solutions of these equations modeling a fluid is difficult. But they often allow, by an approximate resolution, to propose a model of many phenomena, such as ocean currents and air mass movements in the atmosphere for meteorologists, the behavior of skyscrapers or bridges under the action of wind for architects and engineers, or that of airplanes, trains or high-speed cars for their design offices, as well as the flow of water in a pipe and many other flow phenomena of various fluids. In mathematics, nonlinearity complicates things. In physics, too, the difficulty arises. For this term nonlinearity has its translation in the complexity of the physical phenomena described. This difficulty of resolution partly affects the analyses or descriptions of the modeled phenomena. The objective of this work is the search for exact solutions of the Navier-Stokes equations in dimension 2 and in dimension 3. The method of the reduced differential transform is used to find the exact solutions of these Navier-Stokes equations in 2D and 3D. This method gives an algorithm that favors the rapid convergence of the problem to the exact solution sought. Besides the introduction, this article is structured as follows: the presentation of the method, its application on the two selected Navier-Stokes problems whose exact solutions are obtained with ease, then intervenes the conclusion of the whole work.
Abstract: In fluid mechanics, the Navier-Stokes equations are non-linear partial differential equations that describe the motion of Newtonian fluids. A fluid can be a liquid or a gas. Therefore, the Navier-Stokes equation concerns many phenomena that surround us. The analytical resolution, the search for exact solutions of these equations modeling a fluid is...
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Unique Diagram of a Spatial Arc and the Knotting Probability
Issue:
Volume 11, Issue 6, December 2022
Pages:
102-111
Received:
2 November 2022
Accepted:
18 November 2022
Published:
29 November 2022
DOI:
10.11648/j.pamj.20221106.12
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Abstract: There is a question of how a spatial polygonal arc in 3-space such as a linear molecule, a protein, a non-circular DNA, or a linear polymer, etc. is considered as a knot object. This paper answers it by introducing a notion of knotting probability of a spatial arc. As a tool, the knotting probability of an arc diagram which is invariant under isomorphisms of arc diagrams has been already established by the author, which measures how many non-trivial ribbon surface-knots of genus 2 in the 4-space occur when the arc diagram is regarded as a ribbon chord diagram in a 4D research object. A main task of this paper is to show how to obtain an arc diagram uniquely up to isomorphisms from a given oriented spatial polygonal arc. The image of an oriented spatial polygonal arc under the orthogonal projection from the 3-space to a plane along a unit normal vector is not always any arc diagram. It is shown that an arc diagram unique up to isomorphisms which is determined only by the unit normal vector can be obtained by approximating the projection image of an oriented spatial polygonal arc. By combining the resulting arc diagram with the knotting probability of an arc diagram already established, the knotting probability of every spatial polygonal arc belonging to a unit vector is defined. It is also observed that every oriented spatial polygonal arc (except for the case of a polygonal arc contained in a straight line segment) in 3-space admits a unique orthonormal basis of the 3-space. Thus, the knotting probability of a spatial polygonal arc in 3-space is defined. The knotting probabilities of three concrete examples on oriented spatial polygonal arcs are computed.
Abstract: There is a question of how a spatial polygonal arc in 3-space such as a linear molecule, a protein, a non-circular DNA, or a linear polymer, etc. is considered as a knot object. This paper answers it by introducing a notion of knotting probability of a spatial arc. As a tool, the knotting probability of an arc diagram which is invariant under isomo...
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Characterizations and Representations of the Core-EP Inverse and Its Applications
Xianchun Meng,
Ricai Luo,
Xingshou Huang,
Guiying Wang
Issue:
Volume 11, Issue 6, December 2022
Pages:
112-120
Received:
13 October 2022
Accepted:
4 November 2022
Published:
30 November 2022
DOI:
10.11648/j.pamj.20221106.13
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Abstract: Generalized inverse matrices are an important branch of matrix theory, have a wide range of applications in many fields, such as mathematical statistics, system theory, optimization computing and cybernetics etc. This paper mainly studies the correlation properties and applications of the Core-ep inverse. Firstly, we present the characterizations of the Core-EP inverse by the matrix equations, and an example is given for analysis. Secondly, we present a representation for computing the Core-EP inverse, get a representation of Aij⊕ by Cramer rule , and an example is given for analysis. Finally, we study the constrained matrix approximation problem in the Frobenius norm by using the Core-EP inverse: ║Ax-b║F=min subject to x∈R(Ak), where A∈C m,m , we obtain the unique solution to the problem.
Abstract: Generalized inverse matrices are an important branch of matrix theory, have a wide range of applications in many fields, such as mathematical statistics, system theory, optimization computing and cybernetics etc. This paper mainly studies the correlation properties and applications of the Core-ep inverse. Firstly, we present the characterizations o...
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Simulation for Mathematical Modelling of the Effects of a Random Disturbance on Biodiversity of Forestry Biomass
Bazuaye Frank Etin-Osa,
Musa Alex,
Ezeora Jeremiah
Issue:
Volume 11, Issue 6, December 2022
Pages:
121-125
Received:
16 November 2022
Accepted:
5 December 2022
Published:
15 December 2022
DOI:
10.11648/j.pamj.20221106.14
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Abstract: Mathematical modelling and simulation of dynamical system have drawn attention in recent times. In addition, there has been so much interest in dynamical characteristics of population model. Forest biomass system happens to play an important role in population dynamics. The study of forests and forestry preservation has gained tremendous attention in government, among individuals and researchers. The ecological dynamical system is highly vulnerable to random perturbation which can be attributed to the other environmental and climatic factors and other characteristics of the ecosystem. However, there are two factors that may have a high potential to influence the performance of biodiversity gain or loss. Therefore, this study focusses on the effect of a random disturbance of five different degrees on the forestry biomass, the density of the wood-based industry and the density of the synthetic based industry. The ODE 45 solver have been used to tackle this problem. We have utilized the technique of a numerical simulation to predict that a higher random noise perturbation has the potential to predict higher degree of biodiversity gain than a lower random noise. The result indicates that not all random noise driven factors do predict biodiversity loss. These are presented and fully discussed. quantitatively.
Abstract: Mathematical modelling and simulation of dynamical system have drawn attention in recent times. In addition, there has been so much interest in dynamical characteristics of population model. Forest biomass system happens to play an important role in population dynamics. The study of forests and forestry preservation has gained tremendous attention ...
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