scholarly journals Generalized Refinement of Gauss-Seidel Method for Consistently Ordered 2-Cyclic Matrices

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Gashaye Dessalew ◽  
Tesfaye Kebede ◽  
Gurju Awgichew ◽  
Assaye Walelign
Keyword(s):  
Differential Equation ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Rate Of Convergence ◽  
Difference Method ◽  
Linear Equations ◽  
System Of Linear Equations ◽  
Numerical Examples ◽  
Seidel Method ◽  
Minimum Number

This paper presents generalized refinement of Gauss-Seidel method of solving system of linear equations by considering consistently ordered 2-cyclic matrices. Consistently ordered 2-cyclic matrices are obtained while finite difference method is applied to solve differential equation. Suitable theorems are introduced to verify the convergence of this proposed method. To observe the effectiveness of this method, few numerical examples are given. The study points out that, using the generalized refinement of Gauss-Seidel method, we obtain a solution of a problem with a minimum number of iteration and obtain a greater rate of convergence than other previous methods.

Finite Difference Method for a Second-order Ordinary Differential Equation with a Boundary Condition of the Third Kind

2010 ◽  
Vol 10 (1) ◽  
pp. 109-116 ◽  
Author(s):  
P.K. Pandey
Keyword(s):  
Differential Equation ◽  
Boundary Condition ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method ◽  
Second Order ◽  
Point Boundary ◽  
Order Ordinary Differential Equation ◽  
Numerical Examples ◽  
The Third

Abstract We present a second-order finite difference method for obtaining a solution of a second order two-point boundary value problem subject to Sturm's boundary conditions. We use equidistant discretization points, and the discretization of the differential equation at an interior point is based on just two evaluations of the function. Numerical examples are considered and the convergence of the proposed method is proved computationally.

Numerical Dynamics of Nonstandard Finite Difference Method for Nonlinear Delay Differential Equation

2018 ◽  
Vol 28 (11) ◽  
pp. 1850133 ◽  
Author(s):  
Xiaolan Zhuang ◽  
Qi Wang ◽  
Jiechang Wen
Keyword(s):  
Differential Equation ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Delay Differential Equation ◽  
Difference Method ◽  
Discrete System ◽  
Delay Differential ◽  
Nonlinear Delay Differential Equation ◽  
Nonlinear Delay ◽  
Nonstandard Finite Difference

In this paper, we study the dynamics of a nonlinear delay differential equation applied in a nonstandard finite difference method. By analyzing the numerical discrete system, we show that a sequence of Neimark–Sacker bifurcations occur at the equilibrium as the delay increases. Moreover, the existence of local Neimark–Sacker bifurcations is considered, and the direction and stability of periodic solutions bifurcating from the Neimark–Sacker bifurcation of the discrete model are determined by the Neimark–Sacker bifurcation theory of discrete system. Finally, some numerical simulations are adopted to illustrate the corresponding theoretical results.

Use of Finite Difference Method in the Study of Stepped Beams

1998 ◽  
Vol 26 (1) ◽  
pp. 11-24 ◽  
Author(s):  
A. Krishnan ◽  
Geetha George ◽  
P. Malathi
Keyword(s):  
Differential Equation ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Free Vibration ◽  
Difference Method ◽  
Geometric Properties ◽  
Stepped Beam

The analysis of stepped beams using finite difference method normally is carried by use of a single differential equation. Whenever the step is a node, numerical values of average geometric properties are taken for computation. It is expected that, with finer meshes, the solution will converge to an acceptable one. Free vibration studies carried out on a stepped beam do not confirm this expectation. The numerical values converge; but to wrong ones. Some details are presented in this paper.

scholarly journals Application of Fuzzy Finite Difference Scheme for the Non-homogeneous Fuzzy Heat Equation

2021 ◽  
Author(s):  
Samaneh Zabihi ◽  
reza ezzati ◽  
F Fattahzadeh ◽  
J Rashidinia
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Heat Equation ◽  
Difference Scheme ◽  
Difference Method ◽  
Taylor Expansion ◽  
Truncation Error ◽  
Initial Boundary ◽  
Convergence Conditions ◽  
Numerical Examples

Abstract A numerical framework based on fuzzy finite difference is presented for approximating fuzzy triangular solutions of fuzzy partial differential equations by considering the type of $[gH-p]-$differentiability. The fuzzy triangle functions are expanded using full fuzzy Taylor expansion to develop a new fuzzy finite difference method. By considering the type of gH-differentiability, we approximate the fuzzy derivatives with a new fuzzy finite-difference. In particular, we propose using this method to solve non-homogeneous fuzzy heat equation with triangular initial-boundary conditions. We examine the truncation error and the convergence conditions of the proposed method. Several numerical examples are presented to demonstrate the performance of the methods. The final results demonstrate the efficiency and the ability of the new fuzzy finite difference method to produce triangular fuzzy numerical results which are more consistent with existing reality.

scholarly journals Numerical Solution of Stochastic Generalized Fractional Diffusion Equation by Finite Difference Method

2018 ◽  
Vol 23 (4) ◽  
pp. 53 ◽  
Author(s):  
Amaneh Sepahvandzadeh ◽  
Bahman Ghazanfari ◽  
Nader Asadian
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Diffusion Equation ◽  
Numerical Solution ◽  
Difference Method ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Mean Square ◽  
Numerical Examples ◽  
Mean Square Convergence

The present study aimed at solving the stochastic generalized fractional diffusion equation (SGFDE) by means of the random finite difference method (FDM). Moreover, the conditions of mean square convergence of the numerical solution are studied and numerical examples are presented to demonstrate the validity and accuracy of the method.

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