Mohamed El-Gamel

Sinc-Galerkin method for solving linear sixth-order boundary-value problems

mmThere are few techniques available to numerically solve sixth-order boundary-value problems with two-point boundary conditions. In this paper we show that the Sinc-Galerkin method is a very effective tool in numerically solving such problems. The method is then tested on examples with homogeneous and nonhomogeneous boundary conditions and a comparison with the modified decomposition method is made. It is shown that the Sinc-Galerkin method yields better results.

A numerical algorithm for the solution of telegraph equations

Abstract In this paper, we present a new competitive numerical scheme to solve nonlinear telegraph equations. The method is based on Rothe’s approximation in time discretization and on the Wavelet–Galerkin in the spatial discretization. The approximate solutions converge in the space C ( 0 , T ) ; L 2 ( Ω ) ∩ L 2 ( 0 , T ) ; W 0 1 , 2 ( Ω ) to the variational solution. A full error analysis is performed and a numerical experiment is given to illustrate the good convergence behavior of the approximate solution.

On the Galerkin and collocation methods for two-point boundary value problems using sinc bases

This paper presents a study of the performance of the collocation and Galerkin methods using sinc basis functions for solving linear and nonlinear second-order two-point boundary value problems. The two methods have the linear systems solved by the Q-R method and have the nonlinear systems solved by Newtons method. This study shows that the collocation method performs better than the Galerkin method for the cases considered.

Numerical method for the solution of special nonlinear fourth-order boundary value problems

The sinc-Galerkin method is used to approximate solution of nonlinear problems. This work deals with the sinc-Galerkin method for solving nonlinear fourth-order differential equations with homogeneous and nonhomogeneous boundary conditions. The scheme is tested on three nonlinear problems and a comparison with finite difference methods and fourth-order multiderivative method is made. It is shown that the sinc-Galerkin method yields better results.

Sinc and the numerical solution of fifth-order boundary value problems

Sinc methods are a family of self-contained methods of approximation, which have several advantages over classical methods of approximation in the case of the presence of end-point singularities. In this paper we present a fast and accurate numerical scheme for the fifth-order boundary value problems with two-point boundary conditions. The method is then tested on linear and nonlinear examples and a comparison with sixth-degree B-spline functions is made. It is shown that the Sinc-Galerkin method yields better results.

On the numerical solution of linear and nonlinear volterra integral and integro-differential equations

Abstract Sinc bases are developed to approximate the solutions of linear and nonlinear Volterra integral and integro-differential equations. Properties of these sinc bases and some operational matrices are first presented. These properties are then used to reduce the integral and integro-differential equations to systems of linear or nonlinear algebraic equations. Numerical examples illustrate the pertinent features of the method and its applicability to a large variety of problems. The examples include convolution type, singular as well as singularly-perturbed problems.

A Wavelet-Galerkin method for a singularly perturbed convection-dominated diffusion equation

There are few techniques available to numerically solve singularly perturbed parabolic problems. In this paper we show that the Wavelet-Galerkin method is a very effective tool in numerically solving such problems. The method is then tested on several examples and a comparison with the method of reduction order is made. It is shown that the Wavelet-Galerkin method yields better results.

A comparison between the Sinc-Galerkin and the modified decomposition methods for solving two-point boundary-value problems

One of the new techniques used in solving boundary-value problems involving ordinary differential equations is the Sinc-Galerkin method. This method has been shown to be a powerful numerical tool for finding fast and accurate solutions. A less known technique that has been around for almost two decades is the decomposition method. In this paper we solve boundary-value problems of higher order using these two methods and then compare the results. It is shown that the Sinc-Galerkin method in many instances gives better results.

Comparison of the solutions obtained by Adomian decomposition and wavelet-Galerkin methods of boundary-value problems

Abstract In this paper, we will compare the performance of Adomian decomposition method and the wavelet-Galerkin method applied to the solution of boundary-value problems involving non-homogeneous heat and wave equations. It is shown that the Adomian decomposition method in many instances gives better results. In the wavelet-Galerkin solutions, Daubechies six wavelets are used because they give better results than those of lower degree wavelets. The results are then compared with those obtained using the Adomian decomposition method. Although the Adomian decomposition solution required slightly more computational effort than the wavelet-Galerkin solution, it resulted in more accurate results than the wavelet-Galerkin method.

The solution of a time-dependent problem by the B-spline method

Abstract In this paper, we apply the B -spline method for solving the boundary-value problems involving non-homogeneous heat, convection–diffusion, wave and telegraph equations. The accuracy of the method for the equation is O ( ( Δ t ) + h 2 ) . Several examples are given to verify the reliability and efficiency of the method.