Galerkin Method For Numerical Solution Research Articles

Galerkin Method for Numerical Solution of Volterra Integro-Differential Equations with Certain Orthogonal Basis Function

Galerkin Method for Numerical Solution of Volterra Integro-Differential Equations with Certain Orthogonal Basis Function

Cubic B-spline Galerkin method for numerical solution of the coupled nonlinear Schrödinger equation

In this paper, the Galerkin method, based on cubic B-spline function as the shape and weight functions is applied for the numerical solution of the one-dimensional coupled nonlinear Schrödinger equation. Numerical experiments involving single solitary wave, collision of two solitary waves and collision of three solitary waves are conducted. The obtained numerical results of the proposed scheme are compared with the analytical results and previously published numerical results. Two conserved quantities I1 and I2 are calculated for collision of two solitary waves and interaction of three solitary waves. The scheme provides accurate results which are in good agreement when compared to other numerical schemes. The order of convergence of the scheme is calculated. Moreover, the use of cubic B-spline Galerkin method produces smooth solutions without numerical smearing.

Imposing boundary and interface conditions in multi-resolution wavelet Galerkin method for numerical solution of Helmholtz problems

In this paper, we propose a new formulation for numerical solution of boundary value problems using mesh-free multi-resolution wavelet Galerkin (WG) method. A difficulty with the WG method is imposing boundary condition constraints. Single scale wavelet basis functions together with a jump function approach using cubic polynomial functions have been used in solving some boundary problems in the literature. Ramp and cubic polynomial functions have been also used in the jump function approach described in element-free Galerkin context, and it is proven ramp functions exactly satisfy the desirable property for the derivative of jump, but the cubic polynomial functions do not. In this paper, a multi-resolution WG method is employed, and both ramp and cubic polynomial jump functions are applied. Obtained results in one dimension are compared with analytical solutions. Results obtained with ramp jump functions in two dimensions are also reported. The results obtained with the WG method manifest boundary and interface conditions are accurately imposed when the ramp jump functions are applied. Numerical results are in good agreement with accurate solutions.

High Order Accurate Runge Kutta Nodal Discontinuous Galerkin Method for Numerical Solution of Linear Convection Equation

High Order Accurate Runge Kutta Nodal Discontinuous Galerkin Method for Numerical Solution of Linear Convection Equation

Wavelet Galerkin method for numerical solution of nonlinear integral equation

In this paper, the continues Legendre wavelets constructed on the interval [0, 1] are used to solve the nonlinear Volterra and Fredholm integral equation of the second kind.The nonlinear part of the integral equation is approximated by Legendre wavelets, and the nonlinear integral equation is reduced to a system of nonlinear equations. Numerical examples illustrates the pertinent features of the method.

A wavelet-like Galerkin method for numerical solution of variational inequalities arising in elastoplasticity

The well-known problem of elasticity that may be written as a variational equation [4] has been recently extended to non-linear elastoplastic behaviours [13] giving rise to a class of variational inequalities of second kind [9]. This paper presents a wavelet Galerkin method for the numerical solution of such a class of problems, extensively studied and solved in the past by means of more traditional approaches. The novelty of the scheme presented herein is represented by the capability of wavelets to locate regions where plasticity is growing without the need of any further indicator. This is useful for adaptive remeshing as well as for gaining insight into internal variable models with applications to dynamic analysis and damage identification. Copyright © 2000 John Wiley & Sons, Ltd.