Ikram, A. Tirmizi, Fazal-i-Haq and Siraj-ul-Islam,. 2008. Non-polynomial spline solution of singularly perturbed boundary value problems. J. Comput. Appl. Math.
On the Numerical Solution for Singularly Perturbed Second-order ODEs 1
N.H. Sweilam and 2 H.E. Fathy
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2
Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt Department of Mathematics, Faculty of Science, Benha University, Benha, Egypt
Abstract: In this article we consider the approximation of singularly perturbed boundary value problems using a local adaptive grid h-refinement for finite element method, the variation iteration method and the homotopy perturbation method. The solution to such problems contains boundary layers which overlap and interact and the numerical approximation must take this into account in order for the resulting scheme to converge uniformly with respect to the singular perturbation parameters. The results obtained by these methods are compared to the exact solutions for some model problems. It is found that the local adaptive grid h-refinement for finite element method is highly stable methods and always converges to the solution independently on the singular perturbation parameters. Key words: Boundary value problems singularly perturbed problems finite element method adaptive grids homotopy perturbation method variational iteration method •
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INTRODUCTION Consider the singularly perturbed boundary value problem of the form: −e u xx (x) + g(x)u(x) = f(x), x ∈ O = (a,b)
In this work we will solve (1)-(2) using a local adaptive grid h-refinement for finite element method using the recovery based error estimators. The recovery based error estimators first introduce by Zienkiewicz and Zhu in 1987 ([33] and the references therein). These techniques were extensively improved by introducing the new recovery process [33]. We will apply here the Supper Patch Recovery (SPR) method introduced in [34] by the same authors to solve singularly perturbed second-order ordinary differential equations. Using SPR technique we can seek to identify the most reliable and computationally inexpensive variant of the iterative grid refinement, re-refinement algorithm for the finite element solution of singularly perturbed BVPs. The grid refinement is accomplished by adding nodes at inter-nodal positions and rerefinement by removing previously added nodes in reverse order. Re-gridding strategy that uses the mean indicator value as the indicator threshold for refinement [5, 33, 34]. On the other hand, much attention has been devoted recently to numerical methods, which do not require discretization of variables or linearization of the nonlinear equations, among which the variational iteration method (VIM) and the homotopy perturbation method [13-15, 25-27]. The basic idea of variational iteration method is to construct a correction functional with a general Lagrange multiplier which can be identified optimally via variational theory. Also the basic idea of the homotopy perturbation method is to construct a homotopy functional depending on an
two-point
(1)
with the following Dirichlet conditions: u(a) = u(b) = 0
