Department of Mathematics
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Item A uniformly convergent quadratic B-spline collocation method for singularly perturbed parabolic partial differential equations with two small parameters(Springer, 2020-10) Kumar, DevendraThis 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.Item A parameter-uniform scheme for singularly perturbed partial differential equations with a time lag(Wiley, 2019-12) Kumar, DevendraA 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.Item Spline-based parameter-uniform scheme for fourth-order singularly perturbed differential equations(Springer, 2022-08) Kumar, DevendraThis 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.Item Wavelet-based approximation with nonstandard finite difference scheme for singularly perturbed partial integrodifferential equations(Springer, 2022-10) Kumar, DevendraA 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.Item A second-order numerical scheme for the time-fractional partial differential equations with a time delay(Springer, 2022-03) Kumar, DevendraThis 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.Item Second-order convergent scheme for time-fractional partial differential equations with a delay in time(Springer, 2022-10) Kumar, DevendraThis 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.