The Galerkin method for singularly perturbed boundary value problems on adaptive grids

1991 ◽  
Vol 31 (5) ◽  
pp. 817-826 ◽  
Author(s):  
V. V. Strygin ◽  
V. V. Sirunyan
Keyword(s):  
Galerkin Method ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Singularly Perturbed ◽  
Adaptive Grids ◽  
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 Numerical solution of singularly perturbed self-adjoint boundary value problem using Galerkin method

2020 ◽  
Vol 12 (3) ◽  
pp. 26-37
Author(s):  
Murad Hussein Salih ◽  
Gemechis File Duressa ◽  
Habtamu Garoma Debela
Keyword(s):  
Galerkin Method ◽  
Numerical Solution ◽  
Boundary Value Problems ◽  
Basis Function ◽  
Present Method ◽  
Weighting Function ◽  
Boundary Value ◽  
Singularly Perturbed ◽  
Algebraic Equations ◽  
The Given

This paper presents numerical solution of second order singularly perturbed self-adjoint boundary value problems using weighted residual method of Galerkin type. First, for the given problem, the residue was computed using appropriate approximated basis function which satisfies all the boundary conditions. Then, using the chosen weighting function, integrating the weighted residue over the domain and the given differential equation is transformed to linear systems of algebraic equations. Further, these algebraic equations were solved using Galerkin method. To validate the applicability of the proposed method, two model examples have been considered and solved for different values of perturbation parameter and with different order of basis function. Additionally, convergence of error bounds has been established for the method. As it can be observed from the numerical results, the present method approximates the exact solution very well. Moreover, the present method gives better accuracy when the order of basis function is increased and it also improves the result of the methods existing in the literature. Keywords: Singularly perturbed problems, Self-adjoint problem, Galerkin method, Boundary value problems.

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.

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