Department of Mathematics

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Author: search.filters.author.Kumar, DevendraSubject: search.filters.subject.Boundary layers

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Now showing 1 - 7 of 7
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    Trigonometric B-spline based ε-uniform scheme for singularly perturbed problems with Robin boundary conditions
    (Taylor & Francis, 2022-07) Kumar, Devendra
    In this paper, a non-polynomial-based trigonometric cubic B-spline collocation method is developed to solve the reaction-diffusion singularly perturbed problems with Robin boundary conditions. These problems are more tedious to solve than those with Dirichlet and Neumann boundary conditions. The parameter ε in the differential equation results in a rapid change in the solution over a small region. A piecewise uniform mesh is constructed to handle this difficulty. Also, a modification of the proposed mesh is suggested to improve the accuracy of the numerical results by introducing a change in the transition parameter. Through rigorous analysis, it has been shown that the method is almost second-order uniformly convergent. The performance and theoretical findings of the proposed scheme are validated through numerical experiments presented for two test problems. The accuracy of the method is measured in the discrete maximum norm. The tabular results demonstrate that the newly added mesh produces better results.
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    Trigonometric quintic B-spline collocation method for singularly perturbed turning point boundary value problems
    (Taylor & Francis, 2020-08) Kumar, Devendra
    A trigonometric quintic B-spline method is proposed for the solution of a class of turning point singularly perturbed boundary value problems (SP-BVPs) whose solution exhibits either twin boundary layers near both endpoints of the interval of consideration or an interior layer near the turning point. To resolve the boundary/interior layer(s) trigonometric quintic B-spline basis functions are used with a piecewise-uniform mesh generated with the help of a transition parameter that separates the layer and regular regions. The proposed method reduces the problem into a system of algebraic equations which can be written in matrix form with the penta-diagonal coefficient matrix. The well-known fast penta-diagonal system solver algorithm is used to solve the system. The method is shown almost fourth-order convergent irrespective of the size of the diffusion parameter ϵ. The theoretical error bounds are verified by taking some relevant test examples computationally.
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    Parameter-uniform numerical treatment of singularly perturbed initial-boundary value problems with large delay
    (Elsevier, 2020-07) Kumar, Devendra
    In this article, a parameter-uniform implicit scheme is constructed for a class of parabolic singularly perturbed reaction-diffusion initial-boundary value problems with large delay in the spatial direction. In general, the solution of these problems exhibits twin boundary layers and an interior layer (due to the presence of the delay in the reaction term). Crank-Nicolson difference formula (on a uniform mesh) is used in time to semi-discretize the given PDE, and then the standard finite difference scheme (on a piecewise-uniform mesh) is used for the system of ordinary differential equations obtained in the semi-discretization. The convergence analysis shows that the method is ε-uniformly convergent of order two in the temporal direction and almost first-order in the spatial direction. Two test examples are encountered to show the efficiency of the method, validate the computational results, and to confirm the predicted theory.
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    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.
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    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.
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    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.
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    Uniformly convergent scheme for fourth-order singularly perturbed convection-diffusion ODE
    (Elsevier, 2023-04) Kumar, Devendra
    This paper contemplates a numerical investigation of the convection-diffusion type's fourth-order singularly perturbed linear and nonlinear boundary value problems. First, the considered linear fourth-order differential equation is converted into a strongly/weakly coupled singularly perturbed system (depending on the coefficient of the first-order derivative) of two ordinary differential equations with Dirichlet boundary conditions to solve the problem numerically. One of the equations is free from the perturbation parameter in the system. To obtain the solution for this system, we propose a numerical method of quadratic -splines on an exponentially graded mesh. Convergence analysis shows that the proposed numerical scheme is second-order uniformly convergent in the discrete maximum norm. The nonlinear differential equation is linearized using the quasilinearization technique, and then the proposed approach is applied to the linearized problem. The theoretical outcomes are validated by executing the proposed method on three test problems.