AN INITIAL VALUE TECHNIQUE FOR SINGULARLY PERTURBED REACTION-DIFFUSION PROBLEMS WITH A NEGATIVE SHIFT

Abstract. In this paper, a numerical method named as Initial Value Technique (IVT) is suggested to solve singularly perturbed boundary value problems for second order ordinary differential equations of reactiondiffusion type with a delay (negative shift). In this technique, the original problem of solving the second order differential equation is reduced to solving four first order singularly perturbed differential equations without delay and one algebraic equation with a delay. The singularly perturbed problems are solved by a second order hybrid finite difference scheme. An error estimate is derived by using supremum norm and it is of order O(ε + N −2 ln 2 N ), where N is a discretization parameter and ε is the perturbation parameter. Numerical results are provided to illustrate the theoretical results

An hybrid initial value method for singularly perturbed delay differential equations with interior layers and weak boundary layer

In this paper, an hybrid initial value method on Shishkin mesh is suggested to solve singularly perturbed boundary value problem for second order ordinary delay differential equation with discontinuous convection coefficient and source term. In this method, the original problem of solving the second order differential equation is reduced to solving four first order differential equations. Among the four first order differential equations, three of them are singularly perturbed differential equations without delay and other one is a regular differential equation with a delay term. The singularly perturbed differential equations are solved by the second order hybrid finite difference schemes, whereas the delay differential equation is solved by the improved Euler method. An error estimate is derived by using the supremum norm and it is of almost second order convergence. Numerical results are provided to illustrate the theoretical results. Keywords: Singularly perturbed problem, Discontinuous convection coefficient, Shishkin mesh, Dela

Parameter Uniform Numerical Method for Singularly Perturbed 2D Parabolic PDE with Shift in Space

Singularly perturbed 2D parabolic delay differential equations with the discontinuous source term and convection coefficient are taken into consideration in this paper. For the time derivative, we use the fractional implicit Euler method, followed by the fitted finite difference method with bilinear interpolation for locally one-dimensional problems. The proposed method is shown to be almost first-order convergent in the spatial direction and first-order convergent in the temporal direction. Theoretical results are illustrated with numerical examples

Solving a System of One-Dimensional Hyperbolic Delay Differential Equations Using the Method of Lines and Runge-Kutta Methods

In this paper, we consider a system of one-dimensional hyperbolic delay differential equations (HDDEs) and their corresponding initial conditions. HDDEs are a class of differential equations that involve a delay term, which represents the effect of past states on the present state. The delay term poses a challenge for the application of standard numerical methods, which usually require the evaluation of the differential equation at the current step. To overcome this challenge, various numerical methods and analytical techniques have been developed specifically for solving a system of first-order HDDEs. In this study, we investigate these challenges and present some analytical results, such as the maximum principle and stability conditions. Moreover, we examine the propagation of discontinuities in the solution, which provides a comprehensive framework for understanding its behavior. To solve this problem, we employ the method of lines, which is a technique that converts a partial differential equation into a system of ordinary differential equations (ODEs). We then use the Runge–Kutta method, which is a numerical scheme that solves ODEs with high accuracy and stability. We prove the stability and convergence of our method, and we show that the error of our solution is of the order O(Δt+h¯4), where Δt is the time step and h¯ is the average spatial step. We also conduct numerical experiments to validate and evaluate the performance of our method

Streamline diffusion finite element method for singularly perturbed 1D-parabolic convection diffusion differential equations with line discontinuous source

This article presents a study on singularly perturbed 1D parabolic Dirichlet’s type differential equations with discontinuous source terms on an interior line. The time derivative is discretized using the Euler backward method, followed by the application of the streamline–diffusion finite element method (SDFEM) to solve locally one-dimensional stationary problems on a Shishkin mesh. Our proposed method is shown to achieve first-order convergence in time and second-order convergence in space. Our proposed method offers several advantages over existing techniques, including more accurate approximations of the solution on the boundary layer region, better efficiency, and robustness in dealing with discontinuous line source terms. The numerical examples presented in this paper demonstrate the effectiveness and efficiency of our method, which has practical applications in various fields, such as engineering and applied mathematics. Overall, our proposed method provides an effective and efficient solution to the challenging problem of solving singularly perturbed parabolic differential equations with discontinuous line source terms, making it a valuable tool for researchers and practitioners in various domains.Published versio