This paper deals with numerical treatment of singularly perturbed parabolic differential equations having large time delay. The highest order derivative term in the equation is multiplied by a perturbation parameter ε , taking arbitrary value in the interval 0 , 1 . For small values of ε , solution of the problem exhibits an exponential boundary layer on the right side of the spatial domain. The properties and bounds of the solution and its derivatives are discussed. The considered singularly perturbed time delay problem is solved using the Crank-Nicolson method in temporal discretization and exponentially fitted operator finite difference method in spatial discretization. The stability of the scheme is investigated and analysed using comparison principle and solution bound. The uniform convergence of the scheme is discussed and proven. The formulated scheme converges uniformly with linear order of convergence. The theoretical analysis of the scheme is validated by considering numerical test examples for different values of ε .

In this paper, different numerical schemes are developed and investigated for solving singularly perturbed delay differential equations (DDEs). The solution of the considered equations exhibits a strong boundary layer when the perturbation parameter approaches zero. Standard numerical methods developed for solving regular problems fail to treat accurately the considered equations. We developed three numerical schemes for treating the considered equations. The first scheme (Scheme I) uses a non-standard mid-point upwind finite difference method on uniform mesh; the second scheme (Scheme II) uses a standard mid-point upwind finite difference method on Shiskin mesh and the third scheme (Scheme III) uses a non-standard mid-point upwind finite difference method on Shiskin mesh. The existence of a unique discrete solution is investigated using the discrete maximum principle. The stability and uniform convergence of the schemes are investigated and proved. Scheme III gives better accuracy and order of convergence than Scheme I. It has a boundary layer resolving behaviour which is the main drawback of Scheme II. Numerical examples are considered and treated to validate the theoretical finding of the schemes for different values of the perturbation parameter and delay parameter.

Objectives: Numerical treatment of singularly perturbed parabolic delay differential equation is considered. Solution of the equation exhibits a boundary layer, which makes it difficult for numerical computation. Accurate numerical scheme is proposed using $$\theta$$θ -method in time discretization and non-standard finite difference method in space discretization.Result: Stability and uniform convergence of the proposed scheme is investigated. The scheme is uniformly convergent with linear order of convergence before Richardson extrapolation and second order convergent after Richardson extrapolation. Numerical examples are considered to validate the theoretical findings.

In this study, we focus on the formulation and analysis of an exponentially fitted numerical scheme by decomposing the domain into subdomains to solve singularly perturbed differential equations with large negative shift. The solution of problem exhibits twin boundary layers due to the presence of the perturbation parameter and strong interior layer due to the large negative shift. The original domain is divided into six subdomains, such as two boundary layer regions, two interior (interfacing) layer regions, and two regular regions. Constructing an exponentially fitted numerical scheme on each boundary and interior layer subdomains and combining with the solutions on the regular subdomains, we obtain a second order ε‐uniformly convergent numerical scheme. To demonstrate the theoretical results, numerical examples are provided and analyzed.