93 research outputs found

    Numerical solution of singularly perturbed convection–diffusion problem using parameter uniform B-spline collocation method

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    AbstractThis paper is concerned with a numerical scheme to solve a singularly perturbed convection–diffusion problem. The solution of this problem exhibits the boundary layer on the right-hand side of the domain due to the presence of singular perturbation parameter ɛ. The scheme involves B-spline collocation method and appropriate piecewise-uniform Shishkin mesh. Bounds are established for the derivative of the analytical solution. Moreover, the present method is boundary layer resolving as well as second-order uniformly convergent in the maximum norm. A comprehensive analysis has been given to prove the uniform convergence with respect to singular perturbation parameter. Several numerical examples are also given to demonstrate the efficiency of B-spline collocation method and to validate the theoretical aspects

    On Ɛ-uniform convergence of exponentially fitted methods

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    A class of methods constructed to numerically approximate solution of two-point singularly perturbed boundary value problems of the form varepsilonu2˘72˘7+bu2˘7+cu=fvarepsilon u\u27\u27 + b u\u27 + c u = f use exponentials to mimic exponential behavior of the solution in the boundary layer(s). We refer to them as exponentially fitted methods. Such methods are usually exact on polynomials of certain degree and some exponential functions. Shortly, they are exact on exponential sums. It is often possible that consistency of the method follows from the convergence of interpolating function standing behind the method. Because of that, we consider interpolation error for exponential sums. A main result of the paper is an error bound for interpolation by exponential sum to the solution of singularly perturbed problem that does not depend on perturbation parameter varepsilonvarepsilon when varepsilonvarepsilon is small with the respect to mesh width. Numerical experiment implies that the use of dense mesh in the boundary layer for small meshwidth results with varepsilonvarepsilon-uniform convergence

    On Ɛ-uniform convergence of exponentially fitted methods

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    A class of methods constructed to numerically approximate solution of two-point singularly perturbed boundary value problems of the form varepsilonu2˘72˘7+bu2˘7+cu=fvarepsilon u\u27\u27 + b u\u27 + c u = f use exponentials to mimic exponential behavior of the solution in the boundary layer(s). We refer to them as exponentially fitted methods. Such methods are usually exact on polynomials of certain degree and some exponential functions. Shortly, they are exact on exponential sums. It is often possible that consistency of the method follows from the convergence of interpolating function standing behind the method. Because of that, we consider interpolation error for exponential sums. A main result of the paper is an error bound for interpolation by exponential sum to the solution of singularly perturbed problem that does not depend on perturbation parameter varepsilonvarepsilon when varepsilonvarepsilon is small with the respect to mesh width. Numerical experiment implies that the use of dense mesh in the boundary layer for small meshwidth results with varepsilonvarepsilon-uniform convergence

    Numerical solution of singularly perturbed problems using Haar wavelet collocation method

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    Abstract: In this paper, a collocation method based on Haar wavelets is proposed for the numerical solutions of singularly perturbed boundary value problems. The properties of the Haar wavelet expansions together with operational matrix of integration are utilized to convert the problems into systems of algebraic equations with unknown coefficients. To demonstrate the effectiveness and efficiency of the method various benchmark problems are implemented and the comparisons are given with other methods existing in the recent literature. The demonstrated results confirm that the proposed method is considerably efficient, accurate, simple, and computationally attractive
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