Chebyshev-Fourier Spectral Methods for Nonperiodic Boundary Value Problems

A new class of spectral methods for solving two-point boundary value problems for linear ordinary differential equations is presented in the paper. Although these methods are based on trigonometric functions, they can be used for solving periodic as well as nonperiodic problems. Instead of using basis functions periodic on a given interval−1,1, we use functions periodic on a wider interval. The numerical solution of the given problem is sought in terms of the half-range Chebyshev-Fourier (HCF) series, a reorganization of the classical Fourier series using half-range Chebyshev polynomials of the first and second kind which were first introduced by Huybrechs (2010) and further analyzed by Orel and Perne (2012). The numerical solution is constructed as a HCF series via differentiation and multiplication matrices. Moreover, the construction of the method, error analysis, convergence results, and some numerical examples are presented in the paper. The decay of the maximal absolute error according to the truncation numberNfor the new class of Chebyshev-Fourier-collocation (CFC) methods is compared to the decay of the error for the standard class of Chebyshev-collocation (CC) methods.

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Numerical solution of singularly perturbed self-adjoint boundary value problem using Galerkin method

International Journal of Engineering Science and Technology ◽

10.4314/ijest.v12i3.3 ◽

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.

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