Papers by Desta Sodano Sheiso
Social Science Research Network, 2024
Social Science Research Network, 2024
Social Science Research Network, 2024
Social Science Research Network, 2024
Social Science Research Network, 2024
This article presents the numerical approximations to solve singularly perturbed parabolic convec... more This article presents the numerical approximations to solve singularly perturbed parabolic convection-diffusion problems (SPPCDP) with discontinuous initial conditions. The scheme uses backward-Euler for temporal derivatives on a uniform mesh and classical upwind finite difference for spatial derivatives on a piecewise-uniform (Shishkin) mesh. This scheme provides almost a first-order convergence solution in space and time variables. The method employs an upwind finite difference operator on a piecewise-uniform mesh to approximate the gap between the analytic function and the parabolic issue solution. Through comprehensive analysis, we explore the stability and accuracy of the proposed scheme, considering its efficacy in addressing challenges posed by singular perturbations and abrupt changes in the solution. The results provide valuable insights into the applicability of the approach for convection-dominated problems with complex initial conditions, contributing to the advancement of numerical methods in this domain. Parameter-uniform error estimates, stability results, and bounds for the truncation errors are all addressed. Finally, numerical experiments are presented to validate our theoretical results.
Asian Journal of Pure and Applied Mathematics
This article presents the Richardson extrapolation techniques for solving singularly perturbed pa... more This article presents the Richardson extrapolation techniques for solving singularly perturbed parabolic convection-diffusion problems with discontinuous initial conditions (DIC). The scheme uses backward-Euler for temporal derivatives on a uniform mesh and classical upwind finite difference method (FDM) for spatial derivatives on a piecewise-uniform (Shishkin) mesh. This scheme provides almost a first-order convergence solution in both space and time variables. The method employs an upwind finite difference operator on a piecewise-uniform mesh to approximate the gap between the analytic function and the parabolic issue solution. The numerical solution's accuracy is improved by using Richardson extrapolation techniques, which raises it from O(N −1 ln N + ∆t) to O(N −2 ln 2 N + ∆t 2) in the discrete maximum norm, where N is the number of spatial mesh intervals, and ∆t is the size of the temporal step size. Parameter-uniform error estimates, stability results, and bounds for the truncation errors are all addressed. Finally, numerical experiments are presented to validate our theoretical results.
IOSR JOURNAL OF MATHEMATICS.
This article presents the numerical approximations to solve singularly perturbed parabolic convec... more This article presents the numerical approximations to solve singularly perturbed parabolic convection-diffusion problems (SPPCDP) with discontinuous initial conditions. The scheme uses backward-Euler for temporal derivatives on a uniform mesh and classical upwind finite difference for spatial derivatives on a piecewise-uniform (Shishkin) mesh. This scheme provides almost a first-order convergence solution in space and time variables. The method employs an upwind finite difference operator on a piecewise-uniform mesh to approximate the gap between the analytic function and the parabolic issue solution. Through comprehensive analysis, we explore the stability and accuracy of the proposed scheme, considering its efficacy in addressing challenges posed by singular perturbations and abrupt changes in the solution. The results provide valuable insights into the applicability of the approach for convection-dominated problems with complex initial conditions, contributing to the advancement of numerical methods in this domain. Parameter-uniform error estimates, stability results, and bounds for the truncation errors are all addressed. Finally, numerical experiments are presented to validate our theoretical results.
International Journal For Multidisciplinary Research
This study presents the Richardson extrapolation techniques for solving singularly perturbed conv... more This study presents the Richardson extrapolation techniques for solving singularly perturbed convection-diffusion problems (SPCDP) with non-local boundary conditions (NLBC). A numerical approach is presented using an upwind finite difference scheme a piecewise-uniform (Shishkin) mesh. To handle the non-local boundary conditions, the trapezoidal rule is applied. The study establishes an error bound for numerical solutions and determines the numerical approximation for scaled derivatives. To enhance convergence and accuracy, we utilize Richardson extrapolation. This elevates accuracy from O N −1 ln N to O N −2 ln 2 N using this technique, where N is the number of mesh intervals. Numerical results are presented to validate the theoretical findings, demonstrating the effectiveness and accuracy of the proposed technique.
Euler’s Method for Solving Logistic Growth Model Using MATLAB, 2022
This paper introduces Euler's explicit method for solving the numerical solution of the populatio... more This paper introduces Euler's explicit method for solving the numerical solution of the population growth model, logistic growth model. The Euler method is a first-order method, which means that the local error (error per step) is proportional to the square of the step size, and the global error (error at a given time) is proportional to the step size. Euler's method is a numerical method that you can use to approximate the solution to an initial value problem with a differential equation that can't be solved using a more traditional method, like the methods we use to solve separable, exact, or linear differential equations. To validate the applicability of the method on the proposed equation, a model example has been solved for different values of parameters. Using this balance law, we can develop the Logistic Model for population growth. For this model, we assume that we add population at a rate proportional to how many are already there. The numerical results in terms of point wise absolute errors presented in tables and graphs show that the present method approximates the exact solution very well. We discuss and explain the solution of logistic growth of population, the kinds of problems that arise in various fields of sciences and engineering. This study aims to solve numerically Euler's method for solving using the Matlab.
Science Journal of Applied Mathematics and Statistics, 2022
Fourier series are a powerful tool in applied mathematics; indeed, their importance is twofold si... more Fourier series are a powerful tool in applied mathematics; indeed, their importance is twofold since Fourier series are used to represent both periodic real functions as well as solutions admitted by linear partial differential equations with assigned initial and boundary conditions. The idea inspiring the introduction of Fourier series is to approximate a regular periodic function, of period T, via a linear superposition of trigonometric functions of the same period T; thus, Fourier polynomials are constructed. They play, in the case of regular periodic real functions, a role analogue to that one of Taylor polynomials when smooth real functions are considered. In this thesis we will study function approximation by FS method. We will make an attempt to approximate square wave function, line function by FS, and line function by Fourier exponential and trigonometric polynomial. DFT will also be used to approximate function values from data set. We compare the accuracy and the error of Fourier approximation with the actual function and we find that the approximate function is very close to the actual function. We also study the solution of 1D heat equation and Laplace equation by Fourier series method. We compare the solution of heat equation obtained by Fourier series with BTCS. We also compare the solution of Laplace equation obtained by Fourier series with Jacobi iterative method. MATLAB codes for each scheme are presented in appendix and results of running the codes give the numerical solution and graphical solution.
International Journal of Systems Science and Applied Mathematics, 2022
FEM is a valuable approximation tool for the solution of Partial Differential Equations when the ... more FEM is a valuable approximation tool for the solution of Partial Differential Equations when the analytical solutions are difficult or impossible to obtain due to complicated geometry or boundary conditions. The Project work involved collecting facts related to WG and DG-FEMs. WG-FEM is a numerical method that was first proposed and analyzed by Wang and Ye (2013) for general second-order elliptic BVPs on triangular and rectangular meshes. DG-FEMs as developed by Cockburn et al. (1970) uses a discontinuous function space to approximate the exact solution of the equations. The comparison and numerical examples demonstrated that WG-FEMs are viable and hold some advantages over DG-FEMs, due to their properties. Numerical examples demonstrated that WGM generates a smaller linear system to solve than the DGMs. WG-FEM have less unknowns, no need for choosing penalty factor and normal flux is continuous across element interfaces compared to DG-FEMs and the implementation of WG-FEMs is easier than that of DG-FEMs based on error and convergence rate. The computations were done by hand and with the help of MATLAB 2021Rb.
Thesis Chapters by Desta Sodano Sheiso
International Journal of Applied Mathematics and Theoretical Physics, 2022
In order to simulate fluid flow, heat transfer, and other related physical phenomena, it is neces... more In order to simulate fluid flow, heat transfer, and other related physical phenomena, it is necessary to describe the associated physics in mathematical terms. Nearly all the physical phenomena of interest to us in this book are governed by principles of conservation and are expressed in terms of partial differential equations expressing these principles. In this research paper, is a summary of conservation equations (Continuity, Momentum, Species, and Energy) that govern the flow of a Newtonian fluid. In particular, this paper studied the solution of two-dimensional (2D) Navier-Stokes (N-S) equations using the finite difference method (FDM), finite element method (FEM), and finite volume method (FVM) on a test problem of Methane combustion in a laminar diffusion flame. First, the computational domain was decomposed into grids in FDM and elements in FEM, later the Navier-Stokes equations, Energy, and, Species conservation equations were solved at the grid points and a MATLAB code has been written to check the consistency, stability, and accuracy for finer meshes. Following this step, the discretized equations for each sub-domain will be developed using the finite difference and finite element method, resolved using an iterative solver-Gauss Seidel technique. The MATLAB code is written for 2D geometries for science and engineering applications. The focus of this research paper is the development of physical models using numerical methods like FDM, and FEM for modeling science and Engineering applications by using Navier-stokes equations, Energy equations, and Species conservation equations.
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Papers by Desta Sodano Sheiso
Thesis Chapters by Desta Sodano Sheiso