Department of Mathematics

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    A conforming virtual element method for parabolic integro-differential equations
    (De Gruyter, 2023-10) Yadav, Sangita
    This article develops and analyses a conforming virtual element scheme for the spatial discretization of parabolic integro-differential equations combined with backward Euler’s scheme for temporal discretization. With the help of Ritz–Voltera and L2 projection operators, optimal a priori error estimates are established. Moreover, several numerical experiments are presented to confirm the computational efficiency of the proposed scheme and validate the theoretical findings.
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    Optimal Error Estimates of Two Mixed Finite Element Methods for Parabolic Integro-Differential Equations with Nonsmooth Initial Data
    (Springer, 2013-05) Yadav, Sangita
    In the first part of this article, a new mixed method is proposed and analyzed for parabolic integro-differential equations (PIDE) with nonsmooth initial data. Compared to the standard mixed method for PIDE, the present method does not bank on a reformulation using a resolvent operator. Based on energy arguments combined with a repeated use of an integral operator and without using parabolic type duality technique, optimal L2-error estimates are derived for semidiscrete approximations, when the initial condition is in L2. Due to the presence of the integral term, it is, further, observed that a negative norm estimate plays a crucial role in our error analysis. Moreover, the proposed analysis follows the spirit of the proof techniques used in deriving optimal error estimates for finite element approximations to PIDE with smooth data and therefore, it unifies both the theories, i.e., one for smooth data and other for nonsmooth data. Finally, we extend the proposed analysis to the standard mixed method for PIDE with rough initial data and provide an optimal error estimate in L2, which improves upon the results available in the literature.
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    An hp-local Discontinuous Galerkin Method for Parabolic Integro-Differential Equations
    (Springer, 2020-06) Yadav, Sangita
    In this article, a priori error bounds are derived for an hp-local discontinuous Galerkin (LDG) approximation to a parabolic integro-differential equation. It is shown that error estimates in L 2-norm of the gradient as well as of the potential are optimal in the discretizing parameter h and suboptimal in the degree of polynomial p. Due to the presence of the integral term, an introduction of an expanded mixed type Ritz-Volterra projection helps us to achieve optimal estimates. Further, it is observed that a negative norm estimate of the gradient plays a crucial role in our convergence analysis. As in the elliptic case, similar results on order of convergence are established for the semidiscrete method after suitably modifying the numerical fluxes. The optimality of these theoretical results is tested in a series of numerical experiments on two dimensional domains.
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    Modelling War Phenomena with an emphasis on Operation Research
    (IEEE, 2021) Pasari, Sumanta
    Different domains of mathematics and informatics have a great impact on a wide range of army needs. Mathematical models when formulated with appropriate parameters can determine an approximate state of war progress. Fields of operations research are of interest to military scientists primarily to recreate war phenomena inventory maintained, using simulations. This paper reviews basic war models such as Lanchester and Guerrilla models and proposes an inventory model with a practical simulation to demonstrate its applicability in modelling food inventory.
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    A uniformly convergent quadratic B-spline collocation method for singularly perturbed parabolic partial differential equations with two small parameters
    (Springer, 2020-10) Kumar, Devendra
    This paper aims to construct a parameters-uniform numerical scheme to solve the singularly perturbed parabolic partial differential equations whose solution exhibits parabolic (or exponential) boundary layers at both the lateral surfaces of the rectangular domain. The method comprises an implicit Euler scheme on a uniform mesh in the temporal direction and the quadratic B-spline collocation scheme on an exponentially graded mesh in the spatial direction. The exponentially graded mesh is generated by choosing an appropriate mesh generating function which adapts the mesh points in the boundary layers appear in the spatial direction. To establish the error estimates the solution is decomposed into its regular and singular components and the error estimates for these components are obtained separately. We prove the parameters-uniform convergence of the proposed numerical scheme and the method is shown to be of O(N−2x+Δt) where Nx denotes the number of mesh points in the space direction and Δt is the mesh step size in the temporal direction. To support the obtained theoretical estimates, two test examples are considered numerically.
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    A parameter-uniform scheme for singularly perturbed partial differential equations with a time lag
    (Wiley, 2019-12) Kumar, Devendra
    A numerical scheme for a class of singularly perturbed delay parabolic partial differential equations which has wide applications in the various branches of science and engineering is suggested. The solution of these problems exhibits a parabolic boundary layer on the lateral side of the rectangular domain which continuously depends on the perturbation parameter. For the small perturbation parameter, the standard numerical schemes for the solution of these problems fail to resolve the boundary layer(s) and the oscillations occur near the boundary layer. Thus, in this paper to resolve the boundary layer the extended cubic B-spline basis functions consisting of a free parameter λ are used on a fitted-mesh. The extended B-splines are the extension of classical B-splines. To find the best value of λ the optimization technique is adopted. The extended cubic B-splines are an advantage over the classical B-splines as for some optimized value of λ the solution obtained by the extended B-splines is better than the solution obtained by classical B-splines. The method is shown to be first-order accurate in t and almost the second-order accurate in x. It is also shown that this method is better than some existing methods. Several test problems are encountered to validate the theoretical results.
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    Spline-based parameter-uniform scheme for fourth-order singularly perturbed differential equations
    (Springer, 2022-08) Kumar, Devendra
    This paper considers a numerical study for the fourth-order singularly perturbed boundary value problems. The associated differential equation is converted into a weakly coupled system of two singularly perturbed ordinary differential equations with Dirichlet boundary conditions to solve the problem numerically. In the system, one of the equations is independent of the perturbation parameter. To solve this system, we present a numerical technique of quadratic B-splines on an exponentially graded mesh. The established results show that the scheme is second-order uniformly convergent in the discrete maximum norm. The theoretical results are validated using the proposed method on two test problems.
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    Wavelet-based approximation with nonstandard finite difference scheme for singularly perturbed partial integrodifferential equations
    (Springer, 2022-10) Kumar, Devendra
    A non-standard finite difference scheme with Haar wavelet basis functions is constructed for the convection–diffusion type singularly perturbed partial integrodifferential equations. The scheme comprises the Crank–Nicolson time semi-discretization followed by the Haar wavelet approximation in the spatial direction. The presence of the perturbation parameter leads to a boundary layer in the solution’s vicinity of x=1. The Shishkin mesh is constructed to resolve the boundary layer. The method is proved to be parameter-uniform convergent of order two in the L2-norm through meticulous error analysis. Compared to the recent methods developed to solve such problems, the present method is a boundary layer resolving, fast, and elegant.
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    A second-order numerical scheme for the time-fractional partial differential equations with a time delay
    (Springer, 2022-03) Kumar, Devendra
    This work proposes a numerical scheme for a class of time-fractional convection–reaction–diffusion problems with a time lag. Time-fractional derivative is considered in the Caputo sense. The numerical scheme comprises the discretization technique given by Crank and Nicolson in the temporal direction and the spline functions with a tension factor are used in the spatial direction. Through the von Neumann stability analysis, the scheme is shown conditionally stable. Moreover, a rigorous convergence analysis is presented through the Fourier series. Two test problems are solved numerically to verify the effectiveness of the proposed numerical scheme.
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    Second-order convergent scheme for time-fractional partial differential equations with a delay in time
    (Springer, 2022-10) Kumar, Devendra
    This paper aims to construct an effective numerical scheme to solve convection-reaction-diffusion problems consisting of time-fractional derivative and delay in time. First, the semi-discretization process is given for the fractional derivative using a finite-difference scheme with second-order accuracy. Then the cubic B-spline collocation method is employed to get the full discretization. We prove that the suggested scheme is conditionally stable and convergent. Two numerical examples are incorporated to verify the effectiveness of the algorithm. Numerical investigations support the proposed method’s accuracy and show that the method solves the problem efficiently.