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 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 A Finite Difference Fictitious Domain Wavelet Method for Solving Dirichlet Boundary Value Problem

2021 ◽  
Vol 14 (3) ◽  
pp. 706-722
Author(s):  
Francis Ohene Boateng ◽  
Joseph Ackora-Prah ◽  
Benedict Barnes ◽  
John Amoah-Mensah
Keyword(s):  
Finite Difference ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Finite Difference Methods ◽  
Dirichlet Boundary Conditions ◽  
Fictitious Domain ◽  
Dirichlet Boundary Value Problem ◽  
Wavelet Method ◽  
Difference Methods

In this paper, we introduce a Finite Difference Fictitious Domain Wavelet Method (FDFDWM) for solving two dimensional (2D) linear elliptic  partial differential equations (PDEs) with Dirichlet boundary conditions on regular geometric domain. The method reduces the 2D PDE into a 1D system of ordinary differential equations and applies a compactly supported wavelet to approximate the solution. The problem is embedded in a fictitious domain to aid the enforcement of the Dirichlet boundary conditions. We present numerical analysis and show that our method yields better approximation to the solution of the Dirichlet problem than traditional methods like the finite element and finite difference methods.

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