-
Research Article
Application of Lagrange Multiplier for Solving Non – Homogenous Differential Equations
Partha Sarathi Basak*
Issue:
Volume 12, Issue 5, October 2024
Pages:
111-117
Received:
1 August 2024
Accepted:
28 August 2024
Published:
11 September 2024
Abstract: The Lagrange Multiplier method has been applied in solving the regular Sturm Liouville (RSL) equation under a boundary condition of the first kind (Dirichlet boundary condition). This method is a very powerful tool for solving the RSL equation and it involves the solution of the RSL equation with the expansion of eigenfunctions into trigonometric series. The efficiency of this approach is emphasized by solving two examples of regular Sturm Liouville problem under homogenous Dirichlet boundary conditions. The methodology is effectively demonstrated, and the results show a high degree of accuracy of the solution in comparison with the exact solution and reasonably fast convergence.
Abstract: The Lagrange Multiplier method has been applied in solving the regular Sturm Liouville (RSL) equation under a boundary condition of the first kind (Dirichlet boundary condition). This method is a very powerful tool for solving the RSL equation and it involves the solution of the RSL equation with the expansion of eigenfunctions into trigonometric s...
Show More
-
Research Article
Stability Analysis of Degenerate Einstein Model of Brownian Motion
Issue:
Volume 12, Issue 5, October 2024
Pages:
118-132
Received:
9 August 2024
Accepted:
5 September 2024
Published:
19 September 2024
DOI:
10.11648/j.ajam.20241205.12
Downloads:
Views:
Abstract: Recent advancements in stochastic processes have uncovered a paradox associated with the Einstein model of Brownian motion of random particles, which diffuse in the media with no boundary . The classical model developed by Einstein provide diffusion coefficient which does not depend on numbers of particles(concentration) and does not degenerate. Based on this model one can predict the propagation speed of particles movement, conflicting with the second law of thermodynamics. We justify that within Einstein paradigm this issue can be resolved. For that we revisited approach proposed by Einstein, and significantly modified his ideas by introducing inverse Kolmogorov equation, with coefficient degenerating as concentration of the particle of interest vanishes. The modified model successfully resolves paradox affiliated to classical Brownian motion model by introducing a concentration-dependent diffusion matrix, establishing a finite propagation speed. Proposed model utilize but of inverse Kolmogorov stochastic parabolic equation and propose sufficient condition (Hypotheses 1.1) for degeneracy of diffusion coefficient, which guarantee finite speed of propagation inside domain of diffusion. This paper outlines the necessary conditions for this property through a counterexample, which provide infinite speed of propagation for the solution of the equation, with diffusion coefficient, which degenerate as concentration vanishes but with lower speed than in (Hypotheses 1.1). The second part focuses on the stability analysis of the solution of the degenerate Einstein model in case when boundary condition are crucial. We considered degenerate Einstein model in the boundary domain with Dirichlet boundary conditions. Our model bridge degenerate Brownian equation in the bulk of media with boundary of the domain. We with detail investigate stability of the problem with perturbed boundary Data, which vanishes with time. A functional dependence is introduced on the solution that satisfies a specific ordinary differential inequality. The investigation explores the solution's dependence on the boundary and initial data of the original problem, demonstrating asymptotic stability under various conditions. These results have practical applications in understanding stochastic processes and its dependence on the boundary Data within bounded domains.
Abstract: Recent advancements in stochastic processes have uncovered a paradox associated with the Einstein model of Brownian motion of random particles, which diffuse in the media with no boundary . The classical model developed by Einstein provide diffusion coefficient which does not depend on numbers of particles(concentration) and does not degenerate. Ba...
Show More
-
Review Article
Theoretical Approaches Review on Covariance Based Sem Using Lisrel, Partial Least Based Sem Using Smart PLS and Component Based Sem Using Gesca
Umi Narimawati,
Jonathan Sarwono*
Issue:
Volume 12, Issue 5, October 2024
Pages:
133-140
Received:
7 August 2024
Accepted:
2 September 2024
Published:
20 September 2024
Abstract: The aim of the research is to review theories underlying the Structural Equation Modeling (SEM) procedure based on covariance (CBSEM), partial least square (PLSSEM) and component (GESCA SEM). The methods used are meta-analysis and systematic secondary data search. Results of the study are: First, theories underlying the CBSEM, PLSSEM and GESCA SEM procedures produce different characteristics in each SEM model. CBSEM models consist of two sub models, namely 1) Factor Analysis Model consisting of a) Exploratory Factor Analysis (EFA) which is designed for a situation where the relationship between indicators and latent variables is unknown or unclear; b) Confirmatory Factor Analysis (CFA) which is used for research where the researcher already has knowledge about the structure of the underlying latent variable (construct) and c) Full Latent Variable Model (LV). 2) PLSSEM consists of two sub model, namely reflective and formative models. GESCA SEM consists of structural / inner model and measurement / outer model. Second, the primary characteristics of CBSEM, PLSSEM and GESCA SEM are requirements of the amount of data sample; the sample data origin; and the software used to calculate the data due to the different statistical formulation, namely LISREL, SmartPLS and GSCA Pro Windows. Third, the main differences among the CBSEM, PLS SEM and GESCA SEM are in the uses of the unstandardized regression coefficients (b) versus the standardized regression coefficients (β). Thus, the researchers that are going to use those procedures must consider those three important findings.
Abstract: The aim of the research is to review theories underlying the Structural Equation Modeling (SEM) procedure based on covariance (CBSEM), partial least square (PLSSEM) and component (GESCA SEM). The methods used are meta-analysis and systematic secondary data search. Results of the study are: First, theories underlying the CBSEM, PLSSEM and GESCA SEM ...
Show More
-
Research Article
Numerical Simulation of Gas Flow and Nanoparticle Deposition in the Human Lung
Roni Akter*,
Nilufar Yasmin,
Mahtab Uddin Ahmmed
Issue:
Volume 12, Issue 5, October 2024
Pages:
141-148
Received:
24 June 2024
Accepted:
29 July 2024
Published:
26 September 2024
Abstract: This paper aims to investigate the numerical simulation of breathing for nanoparticle deposition of the human lung. The solid particles in the air that are passing through the human respiratory channels (considered as the 18th generation bronchial tube, which is narrow in diameter and short in length) have an impact on how our lungs exchange gases. In this study, a mathematical model within this respiratory tube of the human lung is taken into consideration. The unsteady Navier- Stokes equation is used to represent the fluid particles, and the equation of continuity is used to represent the nanoparticles. The governing equation is simulated numerically using the finite difference techniques under some assumption of axial symmetry and laminar flow, effectively reducing the problem into two dimensions. Results for velocity variation of air and dust particles have been discovered in this discussion. Effects of parameters like Reynolds number and pulse frequency have also been found. Additionally, results demonstrated that the axial velocity of fluid and particles increases with an increase in Reynolds number and frequency along both the length and diameter of the tube. Later, a comparison between fluid and particles for the velocity profile has been discussed.
Abstract: This paper aims to investigate the numerical simulation of breathing for nanoparticle deposition of the human lung. The solid particles in the air that are passing through the human respiratory channels (considered as the 18th generation bronchial tube, which is narrow in diameter and short in length) have an impact on how our lungs exchange gases....
Show More
-
Research Article
Modeling the Two-Strain Dynamics of COVID-19 in Ghana Using a Logistic Growth Model
Issue:
Volume 12, Issue 5, October 2024
Pages:
149-166
Received:
28 July 2024
Accepted:
16 August 2024
Published:
29 September 2024
Abstract: Through mutation, viruses constantly change, bringing into existence new variants; SARS-CoV-2 is no different. In December 2020, variants with different characteristics that could affect transmissibility and death emerged around the world of which Ghana is not an exception. To address this new phenomenon, a two-strain mathematical model of SARS-CoV-2 was formulated to analyzed the transmission dynamics in Ghana. The disease-free equilibrium was calculated. The basic reproduction number, R0= max{R0A, R0B} = max(0.9957945674, 1.109170840), associated with the model is computed using the next generation matrix operator. The disease-free equilibrium is found to be locally asymptotically stable when both R0A, R0B < 1, but unstable otherwise. In addition to the disease-free, the boundary equilibrium for strain A and strain B was also calculated. Using the Gershgorin’s circle theorem, it was shown that the boundary equilibrium is locally asymptotically stable when both R0A, R0B > 1, but unstable when otherwise. Simulations of the model were carried out. Results indicate that the government should intensify its efforts to vaccinate a larger proportion of the population and also recommends implementing comprehensive control measures, such as the use of face masks, social distancing, and contact tracing, to mitigate the spread of the disease.
Abstract: Through mutation, viruses constantly change, bringing into existence new variants; SARS-CoV-2 is no different. In December 2020, variants with different characteristics that could affect transmissibility and death emerged around the world of which Ghana is not an exception. To address this new phenomenon, a two-strain mathematical model of SARS-CoV...
Show More
-
Research Article
Combinatorial Properties, Invariants and Structures Associated with the Direct Product of Alternating and Cyclic Groups Acting on the Cartesian Product of Two Sets
Issue:
Volume 12, Issue 5, October 2024
Pages:
167-174
Received:
19 August 2024
Accepted:
6 September 2024
Published:
29 September 2024
DOI:
10.11648/j.ajam.20241205.16
Downloads:
Views:
Abstract: In relation to group action, much research has focused on the properties of individual permutation groups acting on both ordered and unordered subsets of a set, particularly within the Alternating group and Cyclic group. However, the action of the direct product of Alternating group and Cyclic group on the Cartesian product of two sets remains largely unexplored, suggesting that some properties of this group action are still undiscovered. This research paper therefore, aims to determine the combinatorial properties - specifically transitivity and primitivity - as well as invariants which includes ranks and subdegrees of this group action. Lemmas, theorems and definitions were utilized to achieve the objectives of study with significant use of the Orbit-Stabilizer theorem and Cauchy-Frobeneus lemma. Therefore in this paper, the results shows that for any value of n ≥ 3, the group action is transitive and imprimitive. Additionally, we found out that when n = 3, the rank is 9 and the corresponding subdegrees are ones repeated nine times that is, 1, 1, 1, 1, 1, 1, 1, 1, 1. Also, for any value of n > 4, the rank is 2n with corresponding subdegrees comprising of n suborbits of size 1 and n suborbits of size (n − 1).
Abstract: In relation to group action, much research has focused on the properties of individual permutation groups acting on both ordered and unordered subsets of a set, particularly within the Alternating group and Cyclic group. However, the action of the direct product of Alternating group and Cyclic group on the Cartesian product of two sets remains larg...
Show More
-
Research Article
Suborbital Graphs of Direct Product of the Symmetric Group Acting on the Cartesian Product of Three Sets
Gikunju David Muriuki*,
Nyaga Lewis Namu
Issue:
Volume 12, Issue 5, October 2024
Pages:
175-182
Received:
26 August 2024
Accepted:
25 September 2024
Published:
18 October 2024
Abstract: The study of suborbital graphs is a key area in group theory for it provides a graphical representation of a group action on a set. Moreover, it helps in understanding the combinatorial structures of the action of a group on a set. In this paper, we construct suborbital graphs based on the group action of the direct product of the symmetric group on Cartesian product of three sets through computation of the ranks and subdegrees of the group action. Suborbital graphs are constructed by the use of Sims theorem. The properties of the suborbital graphs are analyzed. In the study it is proven that the rank of the group action of direct product of the symmetric group acting on the Cartesian product of three sets is 8 for all n ≥ 2 and the suborbits are length 1, (n-1), (n-1), (n-1), (n-1)2, (n-1)2, (n-1)2, (n-1)3. We show that the suborbits of the group action are self-paired. Furthermore, it is demostrated that each graph has a girth of 3 for all n > 2 and suborbital graphs of the group action are undirected. It is shown that graphs Γ2 and Γ3 are regular of degree n-1, graphs Γ4, Γ5 and Γ6 of degree (n-1)2 and graph Γ7 is regular of degree (n-1)3. The suborbital graphs Γi(i=1, 2,…, 6) are disconnected, with the number of connected components equal to n2 while suborbital graph Γ7 is connected for all n > 2.
Abstract: The study of suborbital graphs is a key area in group theory for it provides a graphical representation of a group action on a set. Moreover, it helps in understanding the combinatorial structures of the action of a group on a set. In this paper, we construct suborbital graphs based on the group action of the direct product of the symmetric group o...
Show More
-
Methodology Article
Ecological Impact Assessment of Permanent Site of Federal Polytechnic Oko Using Topographic Survey Method
Ayodele Femi Ogundeji,
Ezugwu Charles Nnamdi*
Issue:
Volume 12, Issue 5, October 2024
Pages:
183-199
Received:
29 August 2024
Accepted:
21 September 2024
Published:
18 October 2024
Abstract: Topographic survey based ecological impact assessment provides a mechanism for data integration, development, management and output presentation in a spatial environment. This research involved incorporating spatial data of all salient points at the Permanent Site of Federal Polytechnic, Oko, Anambra state to find a solution to the erosion menace. Differential Global Positioning System (DGPS) receiver was used to acquire spatial data of buildings, roads, spot height, drainages, catchment pits, etc. The topographic data were processed using ArcGIS software to analyze and produce the topographic maps. Presentations from the topographic survey revealed that the catchment topography runs between 160m – 198m height above the datum, the total area of the catchment area being 216320.653m2, perimeter is 2604.449m and the length of the mainstream is 1125.428m. Topographic maps of the area were used to assess the impact of the ecology on the area of study by analyzing the built up area, surface roughness, impervious and pervious surfaces. Hydraulic design of the drainage using best hydraulic section principle for most economic section to carry the flow was determined to solve the erosion problems in the catchment area. The flow rate obtained was 2.55m3/s and dimensions of the channel to be 1.1m depth and 2.2m width. The design of the selected structural members was done to mitigate against the erosion menace in the study area.
Abstract: Topographic survey based ecological impact assessment provides a mechanism for data integration, development, management and output presentation in a spatial environment. This research involved incorporating spatial data of all salient points at the Permanent Site of Federal Polytechnic, Oko, Anambra state to find a solution to the erosion menace. ...
Show More
