A numerical scheme to solve boundary value problems involving singular perturbation

YÜZBAŞI Ş., KARAÇAYIR M.

EUROPEAN JOURNAL OF COMPUTATIONAL MECHANICS, vol.27, no.2, pp.109-122, 2018 (ESCI) identifier identifier

  • Publication Type: Article / Article
  • Volume: 27 Issue: 2
  • Publication Date: 2018
  • Doi Number: 10.1080/17797179.2018.1479552
  • Journal Name: EUROPEAN JOURNAL OF COMPUTATIONAL MECHANICS
  • Journal Indexes: Emerging Sources Citation Index (ESCI), Scopus, Compendex, zbMATH
  • Page Numbers: pp.109-122
  • Keywords: Boundary value problems, ordinary differential equations, Galerkin method, singular perturbation, numerical solutions, GALERKIN-LIKE APPROACH, INTEGRODIFFERENTIAL EQUATIONS, DIFFERENTIAL-EQUATIONS, RESIDUAL CORRECTION, COLLOCATION
  • Akdeniz University Affiliated: Yes

Abstract

In this study, a numerical method is presented in order to approximately solve singularly perturbed second-order differential equations given with boundary conditions. The method uses the set of monomials whose degrees do not exceed a prescribed N as the set of base functions, resulting from the supposition that the approximate solution is a polynomial of degree N whose coefficients are to be determined. Then, following Galerkin's approach, inner product with the base functions are applied to the residual of the approximate solution polynomial. This process, with a suitable incorporation of the boundary conditions, gives rise to an algebraic linear system of size N thorn 1. The approximate polynomial solution is then obtained from the solution of this resulting system. Additionally, a technique, called residual correction, which exploits the linearity of the problem to estimate the error of any computed approximate solution is discussed briefly. The numerical scheme and the residual correction technique are illustrated with two examples.