Department of Mathematics
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Item Trigonometric B-spline based ε-uniform scheme for singularly perturbed problems with Robin boundary conditions(Taylor & Francis, 2022-07) Kumar, DevendraIn 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.Item A uniformly convergent quadratic -spline based scheme for singularly perturbed degenerate parabolic problems(Elsevier, 2022-05) Kumar, DevendraIn this article, a numerical scheme is developed to solve singularly perturbed convection–diffusion type degenerate parabolic problems. The degenerative nature of the problem is due to the coefficient of the convection term. As the perturbation parameter approaches zero, the solution to this problem exhibits a parabolic boundary layer in the neighborhood of the left end side of the domain. The problem is semi-discretized using the Crank–Nicolson scheme, and then the quadratic spline basis functions are used to discretize the semi-discrete problem. A priori bounds for the solution (and its derivatives) of the continuous problem are given, which are necessary to analyze the error. A rigorous error analysis shows that the proposed method is boundary layer resolving and second-order parameter uniformly convergent. Some numerical experiments have been devised to support the theoretical findings and the effectiveness of the proposed scheme.Item Parameter-uniform fitted operator B-spline collocation method for self-adjoint singularly perturbed two-point boundary value problems(ETNA, 2008) Kumar, DevendraIn 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.Item Uniformly convergent scheme for fourth-order singularly perturbed convection-diffusion ODE(Elsevier, 2023-04) Kumar, DevendraThis 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.Item Parameter independent scheme for singularly perturbed problems including a boundary turning point of multiplicity ≥ 1(JAAC, 2022) Kumar, DevendraA numerical scheme is developed for parabolic singularly perturbed boundary value problems, including multiple boundary turning points at the left endpoint of the spatial direction. The highest order derivative of these problems is multiplied by a small parameter , and when it is close to zero, the solution exhibits a parabolic type boundary layer near the left lateral surface of the domain of consideration. Thus, large oscillations appear when classical/standard numerical methods are used to solve the problem, and one cannot achieve the expected accuracy. Thus, the Crank-Nicolson scheme on a uniform mesh in the temporal direction and an upwind scheme on a Shishkin-type mesh in the spatial direction is constructed. The theoretical analysis shows that the method converges irrespective of the size of with accuracy . Three test examples are presented to verify that the computational results agree with the theoretical ones.Item An efficient parameter uniform spline-based technique for singularly perturbed weakly coupled reaction-diffusion systems(Authero, 2022-07) Kumar, DevendraA parameter-uniform numerical scheme for a system of weakly coupled singularly perturbed reaction diffusion equations of arbitrary size with appropriate boundary conditions is investigated. More precisely, quadratic B-spline basis functions with an exponentially graded mesh are used to solve a ` × ` system whose solution exhibits parabolic (or exponential) boundary layers at both endpoints of the domain. A suitable mesh generating function is used to generate the exponentially graded mesh. The decomposition of the solution into regular and singular components is obtained to provide error estimates. A convergence analysis is addressed, which shows a uniform convergence of the second order. To validate the theoretical findings, two test problems are solved numerically