scholarly journals Numerical Solution of Second-Order Fredholm Integrodifferential Equations with Boundary Conditions by Quadrature-Difference Method

2017 ◽  
Vol 2017 ◽  
pp. 1-5
Author(s):  
Chriscella Jalius ◽  
Zanariah Abdul Majid
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Boundary Conditions ◽  
Difference Method ◽  
Numerical Experiments ◽  
Second Order ◽  
Integrodifferential Equations ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Gauss Elimination

In this research, the quadrature-difference method with Gauss Elimination (GE) method is applied for solving the second-order of linear Fredholm integrodifferential equations (LFIDEs). In order to derive an approximation equation, the combinations of Composite Simpson’s 1/3 rule and second-order finite-difference method are used to discretize the second-order of LFIDEs. This approximation equation will be used to generate a system of linear algebraic equations and will be solved by using Gauss Elimination. In addition, the formulation and the implementation of the quadrature-difference method are explained in detail. Finally, some numerical experiments were carried out to examine the accuracy of the proposed method.

A Novel Finite Difference Method for Flow Calculation on Colocated Grids

2008 ◽  
Author(s):  
Z. Z. Xia ◽  
P. Zhang ◽  
R. Z. Wang
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method ◽  
Numerical Solutions ◽  
Fluid Dynamic ◽  
Algebraic Equations ◽  
Duct Flow ◽  
Staggered Grids ◽  
Linear Algebraic Equations ◽  
Incomplete Lu Factorization

A new finite difference method, which removes the need for staggered grids in fluid dynamic computation, is presented. Pressure checker boarding is prevented through a dual-velocity scheme that incorporates the influence of pressure on velocity gradients. A supplementary velocity resulting from the discrete divergence of pressure gradient, together with the main velocity driven by the discretized pressure first-order gradient, is introduced for the discretization of continuity equation. The method in which linear algebraic equations are solved using incomplete LU factorization, removes the pressure-correction equation, and was applied to rectangle duct flow and natural convection in a cubic cavity. These numerical solutions are in excellent agreement with the analytical solutions and those of the algorithm on staggered grids. The new method is shown to be superior in convergence compared to the original one on staggered grids.

scholarly journals On Solution of Fredholm Integrodifferential Equations Using Composite Chebyshev Finite Difference Method

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Z. Pashazadeh Atabakan ◽  
A. Kazemi Nasab ◽  
A. Kılıçman
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Scheme ◽  
Difference Method ◽  
Integrodifferential Equations ◽  
Algebraic Equations ◽  
Block Pulse Functions ◽  
Collocation Points ◽  
Chebyshev Finite Difference Method ◽  
The Given

A new numerical method is introduced for solving linear Fredholm integrodifferential equations which is based on a hybrid of block-pulse functions and Chebyshev polynomials using the well-known Chebyshev-Gauss-Lobatto collocation points. Composite Chebyshev finite difference method is indeed an extension of the Chebyshev finite difference method and can be considered as a nonuniform finite difference scheme. The main advantage of the proposed method is reducing the given problem to a set of algebraic equations. Some examples are given to approve the efficiency and the accuracy of the proposed method.

scholarly journals WO-GRID METHOD OF STRUCTURAL ANALYSIS BASED ON DISCRETE HAAR BASIS. PART 1: ONE-DIMENSIONAL PROBLEMS

2017 ◽  
Vol 13 (4) ◽  
pp. 128-139
Author(s):  
Marina L. Mozgaleva ◽  
Pavel A. Akimov
Keyword(s):  
Structural Analysis ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method ◽  
Grid Method ◽  
Algebraic Equations ◽  
Initial State ◽  
One Dimensional ◽  
Haar Basis ◽  
Linear Algebraic Equations

The distinctive paper is devoted to the two-grid method of structural analysis based on discrete Haar basis (in particular, the simplest one-dimensional problems are under consideration). A brief review of publications of recent years of Russian and foreign specialists devoted to the current trends in the use of wavelet analysis in construction mechanics is given. Approximations of the mesh functions in discrete Haar bases of zero and first levels are described (the mesh function is represented as the sum in which one term is its approximation of the first level, and the second term is so-called complement (up to the initial state) on the grid of the first level). Projectors are constructed for the spaces of vector functions of the original grid to the space of their approximation on the first-level grid and its complement (the detailing component) to the initial state. Basic scheme of the two-grid method is presented. This method allows solution of boundary problems of structural mechanics with the use of matrix operators of significantly smaller dimension. It should be noted that discrete analogue of the initial operator equation (defined on a given interval) is a system of linear algebraic equations (SLAE) constructed within finite difference method (FDM) or the finite element method (FEM). Next, the transition to the resolving SLAE is done. Block Gauss method is used for its direct solution (forward-backward algorithm is realized). We consider a numerical solution of the boundary problem of bending of the Bernoulli beam lying on an elastic foundation (within Winkler model) as a practically important one-dimensional sample. There is good consistency of the results obtained by the proposed method and by standard finite difference method.

FINITE DIFFERENCE METHOD FOR SOLVING THE NONLINEAR DYNAMIC EQUATION OF UNDERWATER TOWED SYSTEM

2014 ◽  
Vol 11 (04) ◽  
pp. 1350060 ◽  
Author(s):  
ZHIJIANG YUAN ◽  
LIANGAN JIN ◽  
WEI CHI ◽  
HENGDOU TIAN
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Numerical Solution ◽  
Difference Method ◽  
Dynamic Equation ◽  
Nonlinear Dynamic ◽  
Computational Time ◽  
Algebraic Equations ◽  
Nonlinear Dynamic Equation ◽  
Nonlinear Dynamic Equations

A wide body of work exists that describes numerical solution for the nonlinear system of underwater towed system. Many researchers usually divide the tow cable with less number elements for the consideration of computational time. However, this type of installation affects the accuracy of the numerical solution. In this paper, a newly finite difference method for solving the nonlinear dynamic equations of the towed system is developed. The mathematical model of tow cable and towed body are both discretized to nonlinear algebraic equations by center finite difference method. A newly discipline for formulating the nonlinear equations and Jacobian matrix of towed system are proposed. We can solve the nonlinear dynamic equation of underwater towed system quickly by using this discipline, when the size of number elements is large.

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