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"Tsega Endalew"
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Integration and peace in East Africa : a history of the Oromo nation
\"This book analyzes the development of indigenous religious, commercial, and political institutions among the Oromo mainly during the relatively peaceful two centuries in its history, from 1704 to 1882. The largest ethnic group in East Africa, the Oromo promoted peace, cultural assimilation, and ethnic integration. This period witnessed flourishing commerce and communication networks that promoted the maturation of Oromo law and government, the integration of foreign ideas, and the assimilation into Oromo of various other cultures\"--Provided by publisher.
Book
Numerical Solution of Two-Dimensional Nonlinear Unsteady Advection-Diffusion-Reaction Equations with Variable Coefficients
The advection-diffusion-reaction (ADR) equation is a fundamental mathematical model used to describe various processes in many different areas of science and engineering. Due to wide applicability of the ADR equation, finding accurate solution is very important to better understand a physical phenomenon represented by the equation. In this study, a numerical scheme for solving two-dimensional unsteady ADR equations with spatially varying velocity and diffusion coefficients is presented. The equations include nonlinear reaction terms. To discretize the ADR equations, the Crank–Nicolson finite difference method is employed with a uniform grid. The resulting nonlinear system of equations is solved using Newton’s method. At each iteration of Newton’s method, the Gauss–Seidel iterative method with sparse matrix computation is utilized to solve the block tridiagonal system and obtain the error correction vector. The consistency and stability of the numerical scheme are investigated. MATLAB codes are developed to implement this combined numerical approach. The validation of the scheme is verified by solving a two-dimensional advection-diffusion equation without reaction term. Numerical tests are provided to show the good performances of the proposed numerical scheme in simulation of ADR problems. The numerical scheme gives accurate results. The obtained numerical solutions are presented graphically. The result of this study may provide insights to apply numerical methods in solving comprehensive models of physical phenomena that capture the underlying situations.
Journal Article
Numerical Simulation of Unsteady 2D Boundary Layer Flow and Heat Transfer Over a Flat Surface Using the Finite Volume Method With a Collocated Grid
In this study, a numerical simulation is conducted to analyze unsteady, two‐dimensional flow and heat transfer over a flat plate using the finite volume method. A fluid with a constant free‐stream temperature and velocity is considered to flow over the flat plate, which is initially warm. The governing equations for mass, momentum, and energy conservation are discretized using a collocated grid framework and solved iteratively. The finite volume method employed exhibits stability and convergence under the prescribed conditions. The development of the hydrodynamic and thermal boundary layers is examined at different time instances. The results show that increasing the Reynolds number leads to thinner velocity boundary layers, while higher Prandtl numbers produce steeper thermal gradients and thinner thermal boundary layers. The transient evolution of flow and temperature fields illustrates the gradual establishment of boundary layers from the leading edge. This study provides valuable insights into the application of the finite volume method for solving unsteady flow and heat transfer problems, demonstrating its effectiveness in capturing transient boundary layer development.
Journal Article
Numerical Solution of Three-Dimensional Transient Heat Conduction Equation in Cylindrical Coordinates
Heat equation is a partial differential equation used to describe the temperature distribution in a heat-conducting body. The implementation of a numerical solution method for heat equation can vary with the geometry of the body. In this study, a three-dimensional transient heat conduction equation was solved by approximating second-order spatial derivatives by five-point central differences in cylindrical coordinates. The stability condition of the numerical method was discussed. A MATLAB code was developed to implement the numerical method. An example was provided in order to demonstrate the method. The numerical solution by the method was in a good agreement with the exact solution for the example considered. The accuracy of the five-point central difference method was compared with that of the three-point central difference method in solving the heat equation in cylindrical coordinates. The solutions obtained by the numerical method in cylindrical coordinates were displayed in the Cartesian coordinate system graphically. The method requires relatively very small time steps for a given mesh spacing to avoid computational instability. The result of this study can provide insights to use appropriate coordinates and more accurate computational methods in solving physical problems described by partial differential equations.
Journal Article
Comparative Study of Finite Difference and Finite Element Methods for Solving 2D Parabolic Convection–Diffusion–Reaction Equations With Variable Coefficients
This study investigates the numerical solution of two‐dimensional parabolic convection–diffusion–reaction (CDR) equations with variable coefficients using the finite difference method (FDM) and the finite element method (FEM). The FDM employs central differences for spatial discretization and the implicit Euler method for time integration, while the FEM uses the Galerkin approach with rectangular elements and three‐point Gauss–Legendre quadrature for spatial integrals, followed by implicit Euler discretization in time. Three test problems are considered to compare accuracy and efficiency. For small diffusion coefficients, the FEM provides higher accuracy, whereas for larger diffusion coefficients, both methods deliver nearly identical accuracy. Despite its improved accuracy in certain cases, the FEM typically involves a higher computational cost than the FDM. Based on the results, the study recommends the use of FEM for problems with boundary or interior layers.
Journal Article
Advancing Finite Difference Solutions for Two‐Dimensional Incompressible Navier–Stokes Equations Using Artificial Compressibility Method and Sparse Matrix Computation
In this article, a numerical scheme for solving two‐dimensional (2D) time‐dependent incompressible Navier–Stokes equations is presented. The artificial compressibility technique is used to incorporate a time derivative of pressure term to the continuity equation. It is employed for pressure–velocity coupling. The scheme consists of backward difference approximation for time derivatives and central difference approximation for spatial derivatives, implemented on a collocated grid. The discretization of the differential equations yields a system of algebraic equations with a block coefficient matrix. To solve this system efficiently, matrix inversion with sparse matrix computation is employed. The proposed numerical scheme is applied to solve three flow problems (lid‐driven cavity flow, rectangular channel flow, and Taylor–Green vortex problem) to validate the accuracy and applicability of the scheme. The results affirm the scheme’s capability to provide precise approximations for solutions to the Navier–Stokes equations. With slight modifications, the scheme can be applied to solve various flow problems with high accuracy, less memory usage, and reduced computational time.
Journal Article
Fitted Operator Method for Singularly Perturbed Delay Parabolic Problems With Boundary Turning Points
In this paper, a numerical scheme for time‐delay singularly perturbed parabolic convection‐diffusion problems with boundary turning points is presented. The solution of the problem shows a steep gradient or rapid variation at the left region of the spatial domain as the perturbation parameter approaches zero. The combination of the singular perturbation parameter, time lag and turning points complicates the problem’s theoretical analysis and numerical solution. Classical numerical methods are inefficient and inaccurate when it comes to solving such complex problems. To overcome these difficulties, first, we discretize the time variable using the Crank–Nicolson method on special uniform mesh such that its discrepancy with time lag lies on the nodal points. Then, we introduce the fitting factor at the diffusion part of the differential equation in order to achieve a uniformly valid solution over the entire region of the domain. The stability and parameter uniform convergence of the scheme are analysed using the minimum principle and solution bounds. It is shown that the scheme is stable and parameter uniform convergence with second‐order accuracy in time and first order in space. Two model examples are presented to demonstrate the scheme’s applicability. Their numerical results reinforce the theoretical analysis.
Journal Article
A Uniformly Convergent Scheme for Singularly Perturbed Unsteady Reaction–Diffusion Problems
In the present work, a class of singularly perturbed unsteady reaction–diffusion problem is considered. With the existence of a small parameter ε , (0 < ε ≪ 1) as a coefficient of the diffusion term in the proposed model problem, there exist twin boundary layer regions near the left end point x = 0 and right end point x = 1 of the spatial domain. The solutions found in such regions have abrupt changes if not fine meshes are used during the spatial domain discretization. Due to these reasons, the classical numerical methods are inefficient to overcome these challenges. To address the suggested problem, we developed and examined a uniformly convergent numerical scheme. The Crank–Nicolson approach for the temporal direction and nonstandard finite difference method for the spatial direction on uniform meshes are used to discretize the continuous problem domain. The uniform convergence analysis shows that the proposed scheme is second‐order uniformly convergent in both temporal and spatial dimensions. Three model examples were given for simulation in order to support the reliability of the formulated scheme, and the obtained results verified that the theoretical analyses coincide with the practical examples. Further, the acquired numerical results demonstrate that the present approach performs better than some of the existing methods in the literature.
Journal Article
A Fitted Linear Multistep Approach for Singularly Perturbed Parabolic‐Type Reaction–Diffusion Problems Using Shishkin Meshes
This paper presents a class of singularly perturbed parabolic‐type reaction diffusion problems. Due to the presence of a small parameter ε , (0 < ε ≪ 1) as a diffusion coefficient, the proposed problem exhibits twin boundary layers in the neighborhood of the end points of the spatial domain near x = 0 and x = 1. The solutions obtained in such layer regions have the properties of oscillations or abrupt changes. Because of these challenges, the classical numerical methods are inefficient to solve the problem. To address these challenges, we formulated and analyzed a parameter uniformly convergent scheme. To approximate the solution, we employ a fitted numerical method combining the implicit Euler scheme for time discretization on a uniform mesh and a linear multistep finite difference scheme for spatial discretization on Shishkin meshes. Richardson extrapolation is used to improve accuracy of numerical computation. The uniform convergence analysis confirmed that the proposed method is uniformly convergent fourth‐order accurate in the spatial direction and second‐order accurate in the temporal direction. Three model examples were presented for simulation in order to verify the applicability of the developed method, and the numerical results validated that the theoretical analyses coincide with the practical examples. Furthermore, the obtained numerical results demonstrate that the new strategy outperforms certain existing methods in the literature.
Journal Article
Hybrid Fitted Mesh Strategy for Singularly Perturbed Time‐Dependent Convection‐Diffusion Problems Featuring Boundary Turning Points
This work investigates the solution of convection‐diffusion parabolic partial‐differential problems with boundary turning points that are singularly perturbed. These types of problems are stiff for the following reason: the small parameter multiplying coefficient of the diffusion term and the presence of boundary turning points. The solution to the problem under consideration in the spatial domain displays a left boundary layer. Analytical or classical numerical approaches confront computing challenges in the rapidly changing solution behaviour in the layer region. To handle this effect, we developed parameter‐uniform numerical method comprised of a hybridized approach that combines central difference and midpoint upwind schemes in space with nonuniform mesh and the Crank–Nicolson method in time with uniform mesh. This scheme is parameter‐uniformly convergent in the maximum norm with second‐order accuracy. Stability is investigated and assessed using the discrete minimum principle and the bounds of truncation error. The numerical solutions of the three model examples considered are aligned with the theoretical conclusions.
Journal Article