New Spectral Second Kind Chebyshev Wavelets Algorithm for Solving Linear and Nonlinear Second-Order Differential Equations Involving Singular and Bratu Type Equations

W M Abd-Elhameed,Y H Youssri,E H Doha

doi:10.1155/2013/715756

Abstract

A new spectral algorithm based on shifted second kind Chebyshev wavelets operational matrices of derivatives is introduced and used for solving linear and nonlinear second-order two-point boundary value problems. The main idea for obtaining spectral numerical solutions for these equations is essentially developed by reducing the linear or nonlinear equations with their initial and/or boundary conditions to a system of linear or nonlinear algebraic equations in the unknown expansion coefficients. Convergence analysis and some efficient specific illustrative examples including singular and Bratu type equations are considered to demonstrate the validity and the applicability of the method. Numerical results obtained are compared favorably with the analytical known solutions.

Highlights

Spectral methods are one of the principal methods of discretization for the numerical solution of differential equations
The main aim of this paper is to develop a new spectral algorithm for solving second-order two-point boundary value problems based on shifted second kind Chebyshev wavelets operational matrix of derivatives
All computations are performed by using Mathematica 8

Summary

Introduction

Spectral methods are one of the principal methods of discretization for the numerical solution of differential equations. The main advantage of these methods lies in their accuracy for a given number of unknowns (see, e.g., [1,2,3]). For smooth problems in simple geometries, they offer exponential rates of convergence/spectral accuracy. Finite difference and finite-element methods yield only algebraic convergence rates. The three most widely used spectral versions are the Galerkin, collocation, and tau methods. Collocation methods have become increasingly popular for solving differential equations, they are very useful in providing highly accurate solutions to nonlinear differential equations (see, e.g., [4, 5])

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