New Wavelets Collocation Method for Solving Second-Order Multipoint Boundary Value Problems Using Chebyshev Polynomials of Third and Fourth Kinds

Abstract

This paper is concerned with introducing two wavelets collocation algorithms for solving linear and nonlinear multipoint boundary value problems. The principal idea for obtaining spectral numerical solutions for such equations is employing third- and fourth-kind Chebyshev wavelets along with the spectral collocation method to transform the differential equation with its boundary conditions to a system of linear or nonlinear algebraic equations in the unknown expansion coefficients which can be efficiently solved. Convergence analysis and some specific numerical examples are discussed to demonstrate the validity and applicability of the proposed algorithms. The obtained numerical results are comparing favorably with the analytical known solutions.

An efficient wavelet based spectral method to singular boundary value problems

“…There is a growing interest in using wavelets to study problems, of greater computational complexity. Wavelet methods have proved to be very effective and efficient tool for solving problems of mathematical calculus [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29]. Among the wavelet transform families the Haar, Legendre wavelets and Chebyshev wavelets deserve much attention [25][26][27][28][29].…”

Some Algorithms for Solving Third-Order Boundary Value Problems Using Novel Operational Matrices of Generalized Jacobi Polynomials

“…In this table, we denote 1 , 2 , 3 , and 4 by the maximum absolute errors if the selected collocation points are, respectively, the zeros of the shifted Legnedre polynomial * +1 ( ), the shifted Chebyshev polynomials of the first and second kinds * +1 ( ) and * +1 ( ), and the shifted nonsymmetirc Jacobi polynomial (1,2) +1 ( ). Moreover, Table 6 displays a comparison between the best errors obtained by the application of GJCOMM with the best errors resulting from the application of the methods mentioned in Example 2.…”

“…The main idea behind spectral methods is to approximate solutions of differential equations by means of truncated series of orthogonal polynomials, say, ∑ . The three popular techniques employed to determine the expansion coefficients are the collocation, tau, and Galerkin methods (see, e.g., [1][2][3]). The collocation approach requires the differential equation to be satisfied exactly at the selected collocation points.…”

Numerical Solution of Time-Fractional Diffusion-Wave Equations via Chebyshev Wavelets Collocation Method

“…Particularly, orthogonal wavelets are widely used in approximating numerical solutions of various types of fractional order differential equations in the relevant literatures; see [18][19][20][21][22]. Among them, the second-kind Chebyshev wavelets have gained much attention due to their useful properties ( [23][24][25][26]) and can handle different types of differential problems. It is observed that most papers using these wavelets methods to approximate numerical solutions of fractional order differential equations are based on the operational matrix of fractional integral or fractional derivatives.…”