scholarly journals On the wavelet-Galerkin method with Deslauriers-Dubuc interpolating scaling functions

2013 ◽  
Vol 37 (2) ◽  
pp. 321-337
Author(s):  
Naohiro Fukuda
Keyword(s):  
Galerkin Method ◽  
Scaling Functions ◽  
Wavelet Galerkin Method

scholarly journals SOLUTION OF 1D AND 2D POISSON'S EQUATION BY USING WAVELET SCALING FUNCTIONS

2016 ◽  
Vol 15 (2) ◽  
pp. 68
Author(s):  
R. B. Burgos ◽  
H. F. C. Peixoto
Keyword(s):  
Galerkin Method ◽  
Poisson’S Equation ◽  
Scaling Functions ◽  
Poisson's Equation ◽  
Linear System Of Equations ◽  
Inner Products ◽  
Connection Coefficients ◽  
Wavelet Finite Element Method ◽  
Stiffness Matrices ◽  
The Galerkin Method

The use of multiresolution techniques and wavelets has become increasingly popular in the development of numerical schemes for the solution of partial differential equations (PDEs). Therefore, the use of wavelet scaling functions as a basis in computational analysis holds some promise due to their compact support, orthogonality and localization properties. Daubechies and Deslauriers-Dubuc functions have been successfully used as basis functions in several schemes like the Wavelet- Galerkin Method (WGM) and the Wavelet Finite Element Method (WFEM). Another possible advantage of their use is the fact that the calculation of integrals of inner products of wavelet scaling functions and their derivatives can be made by solving a linear system of equations, thus avoiding the problem of using approximations by some numerical method. These inner products were defined as connection coefficients and they are employed in the calculation of stiffness matrices and load vectors. In this work, some mathematical foundations regarding wavelet scaling functions, their derivatives and connection coefficients are reviewed. A scheme based on the Galerkin Method is proposed for the direct solution of Poisson's equation (potential problems) in a meshless formulation using interpolating wavelet scaling functions (Interpolets). The applicability of the proposed method and some convergence issues are illustrated by means of a few examples.

scholarly journals Wavelet-Galerkin Quasilinearization Method for Nonlinear Boundary Value Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Umer Saeed ◽  
Mujeeb ur Rehman
Keyword(s):  
Differential Equation ◽  
Galerkin Method ◽  
Boundary Value Problems ◽  
Nonlinear Boundary ◽  
Boundary Value ◽  
Quasilinearization Method ◽  
Nonlinear Boundary Value Problems ◽  
Wavelet Galerkin Method ◽  
Nonlinear Boundary Value ◽  
Quasilinearization Technique

A numerical method is proposed by wavelet-Galerkin and quasilinearization approach for nonlinear boundary value problems. Quasilinearization technique is applied to linearize the nonlinear differential equation and then wavelet-Galerkin method is implemented to linearized differential equations. In each iteration of quasilinearization technique, solution is updated by wavelet-Galerkin method. In order to demonstrate the applicability of proposed method, we consider the various nonlinear boundary value problems.

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