scholarly journals Preconditioning in a wavelet basis and its application to some boundary value problems

2003 ◽  
pp. 86-93
Author(s):  
Marija Rasajski
Keyword(s):  
Finite Difference ◽  
Condition Number ◽  
Iterative Procedure ◽  
Boundary Value ◽  
Finite Difference Methods ◽  
Condition Numbers ◽  
Wavelet Basis ◽  
Numerical Examples ◽  
The Matrix ◽  
Difference Methods

Standard finite difference methods applied to the boundary value problem a(x)u" (x) + b(x)u'(x) + c(x)u(x) = f (x), u(0) = 0, u(1) = 0, lead to linear systems with large condition numbers. Solving a system, i.e. finding the inverse of a matrix with a large condition number can be achieved by some iterative procedure in a large number of iteration steps. By projecting the matrix of the system into the wavelet basis, and applying a diagonal pre-conditioner, we obtain a matrix with a small condition number. Computing the inverse of such a matrix requires fewer iteration steps, and that number does not grow significantly with the size of the system. Numerical examples, with various operators, are presented to illustrate the effect preconditioners have on the condition number, and the number of iteration steps.

scholarly journals Efficient algorithms for solving nonlinear Volterra integro-differential equations with two point boundary conditions

1998 ◽  
Vol 21 (4) ◽  
pp. 755-760 ◽  
Author(s):  
L. E. Garey ◽  
R. E. Shaw
Keyword(s):  
Finite Difference ◽  
Differential Equations ◽  
Iterative Procedure ◽  
Algebraic System ◽  
Finite Difference Methods ◽  
Efficient Algorithms ◽  
Point Boundary ◽  
Nonlinear Case ◽  
Numerical Examples ◽  
Difference Methods

In this paper a collection of efficient algorithms are described for solving an algebraic system with a symmetric Toeplitz coecient matrix. Systems of this form arise when approximating the solution of boundary value Volterra integro-differential equations with finite difference methods. In the nonlinear case, an iterative procedure is required and is incorporated into the algorithms presented. Numerical examples illustrate the results.

scholarly journals ON MATHEMATICAL MODELLING OF METALS DISTRIBUTION IN PEAT LAYERS

2014 ◽  
Vol 19 (4) ◽  
pp. 568-588
Author(s):  
Ilmars Kangro ◽  
Harijs Kalis ◽  
Aigars Gedroics ◽  
Erika Teirumnieka ◽  
Edmunds Teirumnieks
Keyword(s):  
Experimental Data ◽  
Mathematical Model ◽  
Finite Difference ◽  
Diffusion Coefficients ◽  
Boundary Value ◽  
Finite Difference Methods ◽  
Circulant Matrix ◽  
Elliptic Type ◽  
Difference Methods ◽  
The Mathematical Model

In this paper we consider averaging and finite difference methods for solving the 3-D boundary-value problem in multilayered domain. We consider the metals Fe and Ca concentration in the layered peat blocks. Using experimental data the mathematical model for calculation of concentration of metals in different points in peat layers is developed. A specific feature of these problems is that it is necessary to solve the 3-D boundary-value problems for elliptic type partial differential equations (PDEs) of second order with piece-wise diffusion coefficients in the layered domain. We develop here a finite-difference method for solving of a problem of one, two and three peat blocks with periodical boundary condition in x direction. This procedure allows to reduce the 3-D problem to a system of 2-D problems by using circulant matrix.

scholarly journals Stability and Convergence of Two-Step Obrechkoff Scheme For Second-Order Two-Point Boundary Value Problem

2017 ◽  
Vol 4 (1) ◽  
Author(s):  
Olufemi Bosede ◽  
Ashiribo Wusu ◽  
Moses Akanbi
Keyword(s):  
Mathematical Modeling ◽  
Differential Equation ◽  
Finite Difference ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Finite Difference Methods ◽  
Stability And Convergence ◽  
Point Boundary ◽  
Difference Methods ◽  
The Stability

Mathematical modeling of scientific and engineering processes often yield Boundary Value Problems (BVPs). One of the broad categories of numerical methods for solving BVPs is the finite difference methods, in which the differential equation is replaced by a set of difference equations which are solved by direct or iterative methods. In this paper, we use some properties of matrices to analyze the stability and convergence of the prominent finite difference methods - two-step Obrechkoff method - for solving the boundary value problem $u^{\prime \prime} = f(t,u)$, $a < x < b$, $u(a) = \eta_1$, $u(b) = \eta_2$. Conditions for the stability and convergence of the two-step Obrechkoff method method were established.

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