Department of Mathematics

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Author: search.filters.author.Kumar, DevendraSubject: search.filters.subject.Singular perturbation

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Now showing 1 - 8 of 8
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    A parameter-uniform implicit scheme for two-parameter singularly perturbed parabolic problems
    (2023) Kumar, Devendra
    A parameter-uniform implicit approach for two-parameter singularly perturbed boundary valueproblems is constructed. On the solution derivatives, sharp limits are presented. The solution is additionallydivided into regular and singular components, limiting thederivatives of these components utilized in theconvergence analysis. In the temporal direction, the system of ordinary differential equations produced by theCrank-Nicolson scheme on a uniform mesh is further discretized in the spatial direction by employing a finitedifference technique on a selected Shishkin mesh. Through a rigorous analysis, we establish the theoreticalresults for two cases: Case I.ε1/ε22→0 asε2→0, and Case II.ε22/ε1→0 asε1→0, showing that thetechnique is convergent regardless of the magnitude of theε1, ε2parameters. The order of accuracy in Case Iand II are shown to beO((∆t)2+N−1(lnN)2) andO((∆t)2+N−2(lnN)2), respectively. Two examples arepresented to verify the theoretical results
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    A collocation method for singularly perturbed differential-difference turning point problems exhibiting boundary/interior layers
    (Taylor & Francis, 2018-06) Kumar, Devendra
    In this article, a collocation method for the problems where the second-order derivative is multiplied by a small perturbation parameter ϵ, the coefficient of the convection term vanishes at a point within the domain of interest, and the shift δ is of o(ε) is proposed. Similar boundary value problems are encountered while simulating several real-life processes, for instance, first exit time problem in the modelling of neuronal variability. The presence of turning point results into twin boundary layers or an interior layer in the solution of the problem under consideration. A rigorous analysis is carried out and it has been shown theoretically that the numerical solution generated by the method converges uniformly to the solution of the continuous problem with respect to the singular perturbation parameter. The effect of the small shift on the boundary/interior layer(s) has also been observed. Several numerical examples are presented to support the theoretical analysis developed in this article.
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    A parameter-uniform numerical method for time-dependent singularly perturbed differential-difference equations
    (Elsevier, 2011-06) Kumar, Devendra
    A numerical study is made for solving a class of time-dependent singularly perturbed convection–diffusion problems with retarded terms which often arise in computational neuroscience. To approximate the retarded terms, a Taylor’s series expansion has been used and the resulting time-dependent singularly perturbed differential equation is approximated using parameter-uniform numerical methods comprised of a standard implicit finite difference scheme to discretize in the temporal direction on a uniform mesh by means of Rothe’s method and a B-spline collocation method in the spatial direction on a piecewise-uniform mesh of Shishkin type. The method is shown to be accurate of order O(M−1 + N−2 ln3 N), where M and N are the number of mesh points used in the temporal direction and in the spatial direction respectively. An extensive amount of analysis has been carried out to prove the uniform convergence with respect to the singular perturbation parameter. Numerical results are given to illustrate the parameter-uniform convergence of the numerical approximations. Comparisons of the numerical solutions are performed with an upwind and midpoint upwind finite difference scheme on a piecewise-uniform mesh to demonstrate the efficiency of the method.
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    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
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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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    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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    A semi-analytic method for solving singularly perturbed twin-layer problems with a turning point
    (Vilnius Gediminas Technical University, 2023) Kumar, Devendra
    This computational study investigates a class of singularly perturbed second-order boundary-value problems having dual (twin) boundary layers and simple turning points. It is well-known that the classical discretization methods fail to resolve sharp gradients arising in solving singularly perturbed differential equations as the perturbation (diffusion) parameter decreases, i.e., ε → 0+. To this end, this paper proposes a semi-analytic hybrid method consisting of a numerical procedure based on finite differences and an asymptotic method called the Successive Complementary Expansion Method (SCEM) to approximate the solution of such problems. Two numerical experiments are provided to demonstrate the method’s implementation and to evaluate its computational performance. Several comparisons with the numerical results existing in the literature are also made. The numerical observations reveal that the hybrid method leads to good solution profiles and achieves this in only a few iterations.
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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.