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Research Article
Comparative Analysis of Finite Difference and Crank-nicolson Schemes in Simulating Air Pollution Dispersion
Jimrise Ochwach*
,
Julius Njiru Nyaga,
Mark Okongo
Issue:
Volume 15, Issue 4, August 2026
Pages:
123-132
Received:
5 October 2025
Accepted:
18 October 2025
Published:
11 August 2026
DOI:
10.11648/j.acm.20261504.11
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Abstract: Accurate prediction of atmospheric pollutant transport is essential for air quality assessment and environmental management. Classical advection-diffusion models often neglect the combined effects of temperature and humidity, despite their significant influence on pollutant transport and dispersion within the atmospheric boundary layer. This study develops a two-dimensional meteorology-dependent advection-diffusion-reaction model that incorporates temperature-dependent thermodiffusion through the Soret effect and humidity-dependent modification of advective transport. The governing equation is solved numerically using the Explicit Finite Difference (FD) and Crank-Nicolson (CN) schemes to evaluate their accuracy, stability, and computational performance. Von Neumann stability analysis is employed to establish the stability conditions of both numerical methods. The analysis shows that the FD scheme is conditionally stable and requires restrictive time-step selection to satisfy the Courant-Friedrichs-Lewy condition, whereas the CN scheme remains unconditionally stable for the diffusion component and demonstrates superior numerical robustness under practical simulation conditions. Numerical experiments implemented in MATLAB reveal that the CN scheme produces smoother concentration profiles, reduced numerical diffusion, and improved solution accuracy, particularly for long simulation periods. The simulations further indicate that increased wind velocity enhances pollutant transport, higher relative humidity suppresses dispersion and increases pollutant residence time, and elevated temperatures promote stronger atmospheric mixing through thermally induced diffusion. These findings demonstrate that incorporating meteorological variables substantially improves the physical realism of atmospheric dispersion models. The study concludes that the Crank-Nicolson method provides a more reliable and efficient numerical framework for simulating pollutant transport under varying environmental conditions and offers a practical tool for urban air quality assessment, environmental planning, and evidence-based pollution control strategies.
Abstract: Accurate prediction of atmospheric pollutant transport is essential for air quality assessment and environmental management. Classical advection-diffusion models often neglect the combined effects of temperature and humidity, despite their significant influence on pollutant transport and dispersion within the atmospheric boundary layer. This study ...
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Research Article
A Crank-nicolson Finite Element Treatment of Time Singularities of the One Dimensional Heat Equation
Jake Leonard Nkeck*
Issue:
Volume 15, Issue 4, August 2026
Pages:
133-143
Received:
4 June 2026
Accepted:
15 June 2026
Published:
12 August 2026
Abstract: The Heat Equation is a well-known partial differential equation that can be solved numerically by finite difference methods in time coupled with finite element methods in space. Crank-Nicolson methods can therefore be applied as finite difference methods and coupled with linear Lagrange finite element methods in order to solve the Heat equation and obtain an efficient convergence rate, the Heat equation is said to be solved by a Crank-Nicolson finite element method. However, the convergence rate of the Crank-Nicolson finite element method for the Heat equation can be affected if the exact solution entails time singularities; in that case the lack of smoothness of the solution though it is local in time, affects the convergence of the finite element method in the whole domain. This paper presents a Crank-Nicolson finite element method coupled to a Predictor-corrector algorithm to recover the optimal convergence rate when the solution has time singularities. The finite element method presented is based on the approximation of the time singular functions using a Fourier decomposition of the exact solution that leads to computable formulas of the time dependent coefficients of singularities that reduce the smoothness of the solution; so removing those coefficients ameliorate the efficiency of the Crank-Nicolson finite element method. Numerical experiments are presented to show the efficiency of the method.
Abstract: The Heat Equation is a well-known partial differential equation that can be solved numerically by finite difference methods in time coupled with finite element methods in space. Crank-Nicolson methods can therefore be applied as finite difference methods and coupled with linear Lagrange finite element methods in order to solve the Heat equation and...
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Review Article
Qualitative Analysis of Financial Literacy and Loan Repayment Dynamics Among MSMEs in Kenya
Issue:
Volume 15, Issue 4, August 2026
Pages:
144-153
Received:
23 June 2026
Accepted:
6 July 2026
Published:
21 August 2026
Abstract: Micro, Small and Medium Enterprises (MSMEs) play a significant role in employment, income generation and Gross Domestic Product (GDP) in Kenya, but many enterprises remain financially weak, carry heavy loan debt and struggle to repay. This study formulates a nonlinear ordinary differential equation (ODE) model to investigate the interaction between financial literacy, financial health, loan burden, cash reserves and stressors in the business environment for MSMEs in Kenya. The model provides a flexible tool for analysing the equilibrium behavior, positivity, boundedness, local stability and parameter sensitivity. The analysis shows that under economically sensible assumptions the model solutions are non-negative and bounded. An asymptotic stable equilibrium is constructed and calculated to be cash exhausted with the baseline parameter set and found to be locally stable. Sensitivity results show that financial efficacy and literacy intervention have positive effects on financial health, while literacy decay, debt-servicing and business stressors have negative effects on enterprise sustainability. The phase portrait and sensitivity visualization illustrate the financial-health--loan-burden trajectory and the relative influence of the principal model parameters. The study yields a mathematical framework for designing financial literacy programs, credit-management strategies, and interventions to support loan repayment that can help improve the resilience of MSMEs and their ability to repay loans.
Abstract: Micro, Small and Medium Enterprises (MSMEs) play a significant role in employment, income generation and Gross Domestic Product (GDP) in Kenya, but many enterprises remain financially weak, carry heavy loan debt and struggle to repay. This study formulates a nonlinear ordinary differential equation (ODE) model to investigate the interaction between...
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Research Article
The Critical Patch Size Problem in Random Graphs
Issue:
Volume 15, Issue 4, August 2026
Pages:
154-161
Received:
7 April 2026
Accepted:
22 April 2026
Published:
27 August 2026
DOI:
10.11648/j.acm.20261504.14
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Abstract: The problem of critical patch size --- a threshold condition for population persistence --- is investigated in the context of discrete habitats, modeled as graphs with a distinguished subset of vertices acting as sinks. These sinks impose boundary-like constraints analogous to Dirichlet conditions in continuous domains. The population proliferates locally at the vertices and diffuse across the network through the graph Laplacian. In the sinks the population cannot survive. The Dirichlet eigenvalue of the habitat is defined as the smallest eigenvalue of the principal submatrix of the Laplacian obtained by removing the rows and columns associated with sink vertices. This spectral parameter governs the habitat's viability: survival occurs when the Dirichlet eigenvalue of the habitat lies below a critical reaction-to-diffusion ratio. We study survival conditions for a sequence of random habitats built on binomial random graphs. We establish a law of large numbers for the corresponding sequence of Dirichlet eigenvalues and prove the emergence of a sharp threshold phenomenon: with high probability, a large random habitat is either viable or non-viable, depending on whether the reaction-to-diffusion ratio lies below or above this threshold. Our results provide the first general spectral theory for critical patch size on graphs, with implications for ecology, synthetic biology, and the modeling of processes on brain connectomes.
Abstract: The problem of critical patch size --- a threshold condition for population persistence --- is investigated in the context of discrete habitats, modeled as graphs with a distinguished subset of vertices acting as sinks. These sinks impose boundary-like constraints analogous to Dirichlet conditions in continuous domains. The population proliferates ...
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