Error bounds for the Galerkin method applied to singular and nonsingular boundary value problems

1972 ◽  
Vol 19 (3) ◽  
pp. 266-282 ◽  
Author(s):  
J. W. Dailey ◽  
J. G. Pierce
Keyword(s):  
Galerkin Method ◽  
Boundary Value Problems ◽  
Error Bounds ◽  
Boundary Value ◽  
The Galerkin Method

scholarly journals Galerkin Method for Nonlinear Higher-Order Boundary Value Problems Based on Chebyshev Polynomials

2021 ◽  
Vol 2128 (1) ◽  
pp. 012035
Author(s):  
W. Abbas ◽  
Mohamed Fathy ◽  
M. Mostafa ◽  
A. M. A Hesham
Keyword(s):  
Galerkin Method ◽  
Boundary Value Problems ◽  
Chebyshev Polynomials ◽  
Nonlinear Boundary ◽  
Analytic Solution ◽  
Boundary Value ◽  
Higher Order ◽  
Nonlinear Boundary Value Problems ◽  
Algebraic Set ◽  
The Galerkin Method

Abstract In the current paper, we develop an algorithm to approximate the analytic solution for the nonlinear boundary value problems in higher-order based on the Galerkin method. Chebyshev polynomials are introduced as bases of the solution. Meanwhile, some theorems are deducted to simplify the nonlinear algebraic set resulted from applying the Galerkin method, while Newton’s method is used to solve the resulting nonlinear system. Numerous examples are presented to prove the usefulness and effectiveness of this algorithm in comparison with some other methods.

scholarly journals Efficient Solutions of Multidimensional Sixth-Order Boundary Value Problems Using Symmetric Generalized Jacobi-Galerkin Method

2012 ◽  
Vol 2012 ◽  
pp. 1-19 ◽  
Author(s):  
E. H. Doha ◽  
W. M. Abd-Elhameed
Keyword(s):  
Galerkin Method ◽  
Boundary Value Problems ◽  
Jacobi Polynomials ◽  
Boundary Value ◽  
Efficient Solutions ◽  
Sixth Order ◽  
Condition Numbers ◽  
Basic Functions ◽  
The Galerkin Method ◽  
Fourth Order Elliptic Equations

This paper presents some efficient spectral algorithms for solving linear sixth-order two-point boundary value problems in one dimension based on the application of the Galerkin method. The proposed algorithms are extended to solve the two-dimensional sixth-order differential equations. A family of symmetric generalized Jacobi polynomials is introduced and used as basic functions. The algorithms lead to linear systems with specially structured matrices that can be efficiently inverted. The various matrix systems resulting from the proposed algorithms are carefully investigated, especially their condition numbers and their complexities. These algorithms are extensions to some of the algorithms proposed by Doha and Abd-Elhameed (2002) and Doha and Bhrawy (2008) for second- and fourth-order elliptic equations, respectively. Three numerical results are presented to demonstrate the efficiency and the applicability of the proposed algorithms.

scholarly journals New Formulae for the High-Order Derivatives of Some Jacobi Polynomials: An Application to Some High-Order Boundary Value Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-11
Author(s):  
W. M. Abd-Elhameed
Keyword(s):  
Galerkin Method ◽  
Boundary Value Problems ◽  
Chebyshev Polynomials ◽  
Hypergeometric Functions ◽  
Jacobi Polynomials ◽  
Boundary Value ◽  
High Order ◽  
Sixth Order ◽  
Special Cases ◽  
Derivatives Of

This paper is concerned with deriving some new formulae expressing explicitly the high-order derivatives of Jacobi polynomials whose parameters difference is one or two of any degree and of any order in terms of their corresponding Jacobi polynomials. The derivatives formulae for Chebyshev polynomials of third and fourth kinds of any degree and of any order in terms of their corresponding Chebyshev polynomials are deduced as special cases. Some new reduction formulae for summing some terminating hypergeometric functions of unit argument are also deduced. As an application, and with the aid of the new introduced derivatives formulae, an algorithm for solving special sixth-order boundary value problems are implemented with the aid of applying Galerkin method. A numerical example is presented hoping to ascertain the validity and the applicability of the proposed algorithms.

scholarly journals A Galerkin method ofO(h2)for singular boundary value problems

2002 ◽  
Vol 29 (6) ◽  
pp. 361-369
Author(s):  
G. K. Beg ◽  
M. A. El-Gebeily
Keyword(s):  
Galerkin Method ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Basis Functions ◽  
Singular Boundary ◽  
Point Boundary ◽  
Singular Boundary Value Problems ◽  
Special Basis

We describe a Galerkin method with special basis functions for a class of singular two-point boundary value problems. The convergence is shown which is ofO(h2)for a certain subclass of the problems.

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