An Efficient Wavelet-Based Collocation Method for Handling Singularly Perturbed Boundary-Value Problems in Fluid Mechanics

2017 ◽  
Vol 18 (6) ◽  
pp. 485-494
Author(s):  
Firdous A. Shah ◽  
Rustam Abass
Keyword(s):  
Fluid Mechanics ◽  
Boundary Value Problems ◽  
Collocation Method ◽  
Boundary Value ◽  
Operational Matrix ◽  
Haar Wavelet ◽  
Singularly Perturbed ◽  
Test Problems ◽  
Algebraic Equations ◽  
Specific Test

AbstractIn this article, we develop an accurate and efficient wavelet-based collocation method for solving both linear and nonlinear singularly perturbed boundary-value problems that arise in fluid mechanics. The properties of the Haar wavelet expansions together with operational matrix of integration are used to convert the underlying problems into systems of algebraic equations which can be efficiently solved by suitable solvers. The performance of the numerical scheme is assessed and tested on specific test problems and the comparisons are given with other methods existing in the recent literature. The numerical outcomes indicate that the method yields highly accurate results and is computationally more efficient than the existing ones.

Generalized wavelet collocation method for solving fractional relaxation–oscillation equation arising in fluid mechanics

2017 ◽  
Vol 06 (02) ◽  
pp. 1750016
Author(s):  
Firdous A. Shah ◽  
R. Abass
Keyword(s):  
Fluid Mechanics ◽  
Relaxation Oscillation ◽  
Operational Matrix ◽  
Haar Wavelet ◽  
Test Problems ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Specific Test ◽  
Fractional Relaxation ◽  
Oscillation Equation

In this paper, a generalized wavelet collocation operational matrix method based on Haar wavelets is proposed to solve fractional relaxation–oscillation equation arising in fluid mechanics. Contrary to wavelet operational methods accessible in the literature, we derive an explicit form for the Haar wavelet operational matrices of fractional order integration without using the block pulse functions. The properties of the Haar wavelet expansions together with operational matrix of integration are utilized to convert the problems into systems of algebraic equations with unknown coefficients. The performance of the numerical scheme is assessed and tested on specific test problems and the comparisons are given with other methods existing in the recent literature. The numerical outcomes indicate that the method yields highly accurate results and is computationally more efficient than the existing ones.

scholarly journals Numerical Solutions for the Eighth-Order Initial and Boundary Value Problems Using the Second Kind Chebyshev Wavelets

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Xiaoyong Xu ◽  
Fengying Zhou
Keyword(s):  
Boundary Value Problems ◽  
Collocation Method ◽  
Numerical Solutions ◽  
Boundary Value ◽  
Operational Matrix ◽  
Algebraic Equations ◽  
Spline Collocation ◽  
Eighth Order ◽  
Chebyshev Wavelets ◽  
Operational Matrix Of Integration

A collocation method based on the second kind Chebyshev wavelets is proposed for the numerical solution of eighth-order two-point boundary value problems (BVPs) and initial value problems (IVPs) in ordinary differential equations. The second kind Chebyshev wavelets operational matrix of integration is derived and used to transform the problem to a system of algebraic equations. The uniform convergence analysis and error estimation for the proposed method are given. Accuracy and efficiency of the suggested method are established through comparing with the existing quintic B-spline collocation method, homotopy asymptotic method, and modified decomposition method. Numerical results obtained by the present method are in good agreement with the exact solutions available in the literatures.

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

2015 ◽  
Vol 2015 ◽  
pp. 1-10 ◽  
Author(s):  
W. M. Abd-Elhameed
Keyword(s):  
Boundary Value Problems ◽  
Jacobi Polynomials ◽  
Boundary Value ◽  
Operational Matrix ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Third Order ◽  
Generalized Jacobi Polynomials ◽  
Nonlinear Algebraic Equations ◽  
Derivatives Of

The main aim of this research article is to develop two new algorithms for handling linear and nonlinear third-order boundary value problems. For this purpose, a novel operational matrix of derivatives of certain nonsymmetric generalized Jacobi polynomials is established. The suggested algorithms are built on utilizing the Galerkin and collocation spectral methods. Moreover, the principle idea behind these algorithms is based on converting the boundary value problems governed by their boundary conditions into systems of linear or nonlinear algebraic equations which can be efficiently solved by suitable solvers. We support our algorithms by a careful investigation of the convergence analysis of the suggested nonsymmetric generalized Jacobi expansion. Some illustrative examples are given for the sake of indicating the high accuracy and efficiency of the two proposed algorithms.

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