Department of Mathematics

Permanent URI for this collectionhttp://localhost:4000/handle/123456789/1920

Browse

Filters

2007 - 20085

Settings

Author: search.filters.author.Kumar, DevendraEnd date: 2009

Search Results

Now showing 1 - 5 of 5
  • No Thumbnail Available
    Item
    Comparative study of singularly perturbed two-point BVPs via: Fitted-mesh finite difference method, B-spline collocation method and finite element method
    (Elsevier, 2008-10) Kumar, Devendra
    The objective of this paper is to present a comparative study of fitted-mesh finite difference method, B-spline collocation method and finite element method for general singularly perturbed two-point boundary value problems. Due to the small parameter , the boundary layer arises. We have taken a piecewise-uniform fitted-mesh to resolve the boundary layer and we have shown that fitted-mesh finite difference method has -uniform first order convergence, B-spline collocation method has almost second order -uniform convergence and Ritz–Galerkin method
  • No Thumbnail Available
    Item
    Fitted mesh B-spline collocation method for singularly perturbed differential–difference equations with small delay
    (Elsevier, 2008-10) Kumar, Devendra
    This paper deals with the singularly perturbed boundary value problem for a linear second order differential–difference equation of the convection–diffusion type with small delay parameter of whose solution has a boundary layer. The fitted mesh technique is employed to generate a piecewise-uniform mesh, condensed in the neighborhood of the boundary layers. B-spline collocation method is used with fitted mesh. Parameter-uniform convergence analysis of the method is discussed. The method is shown to have almost second order parameter-uniform convergence. The effect of small delay on boundary layer has also been discussed. Several examples are considered to demonstrate the performance of the proposed scheme and how the size of the delay argument and the coefficient of the delay term affects the layer behavior of the solution.
  • No Thumbnail Available
    Item
    A non-linear single step explicit scheme for non-linear two-point singularly perturbed boundary value problems via initial value technique
    (Elsevier, 2008-08) Kumar, Devendra
    In this paper, a method based on initial value technique is proposed for solving non-linear two-point singularly perturbed boundary value problems for second order ordinary differential equations (ODEs) with a boundary layer at one (either left or right) end. The original singularly perturbed boundary value problem is reduced to an initial value problem approximated by its outer solution (asymptotic approximation). The new initial value problem is solved by proposed non-linear single step explicit scheme followed the idea given in [F.D. Van Niekerk, Non-linear one-step methods for initial value problems, Comput. Math. Appl. 13 (1987) 367–371]. The proposed scheme has been shown to be of order two. To demonstrate the applicability of the proposed scheme several (linear and non-linear) problems have been solved. It is observed that the present scheme approximate the exact solution very well.
  • No Thumbnail Available
    Item
    Parameter-uniform fitted operator B-spline collocation method for self-adjoint singularly perturbed two-point boundary value problems
    (ETNA, 2008) Kumar, Devendra
    In this paper, we develop a B-spline collocation method for the numerical solution of a self-adjoint singularly perturbed boundary value problem of the form We construct a fitting factor and use the B-spline collocation method, which leads to a tridiagonal linear system. The method is analyzed for parameter-uniform convergence. Several numerical examples are reported which demonstrate the efficiency of the proposed method.
  • No Thumbnail Available
    Item
    Geometric mesh FDM for self-adjoint singular perturbation boundary value problems
    (Elsevier, 2007-07) Kumar, Devendra
    A numerical method based on finite difference method with variable mesh is given for second order singularly perturbed self-adjoint two point boundary value problems. The original problem is reduced to its normal form and the reduced problem is solved by FDM taking variable mesh(geometric mesh). The maximum absolute errors , for different values of parameter ϵ, number of points N, and the mesh ratio r, for three examples have been given in tables to support the efficiency of the method.