Abstract
This paper deals with numerical treatment of singularly perturbed differential difference equations involving mixed small shifts on the reaction terms. The highest-order derivative term in the equation is multiplied by a small perturbation parameter ε taking arbitrary values in the interval 0,1 . For small values of ε , the solution of the problem exhibits exponential boundary layer on the left or right side of the domain and the derivatives of the solution behave boundlessly large. The terms having the shifts are treated using Taylor’s series approximation. The resulting singularly perturbed boundary value problem is solved using exponentially fitted operator FDM. Uniform stability of the scheme is investigated and analysed using comparison principle and solution bound. The formulated scheme converges uniformly with linear order before Richardson extrapolation and quadratic order after Richardson extrapolation. The theoretical analysis of the scheme is validated using numerical test examples for different values of ε and mesh number N .
Highlights
Differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology
Differential difference equations (DDEs) are differential equations where the evolution of the system depends on the present state of the system and depends on the past history
Perturbed differential difference equations are differential equations in which the highest-order derivative term is multiplied by a small perturbation parameter ε and involves at least one term with delay
Summary
Differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology. When the perturbation parameter tends to zero, the smoothness of the solution of the singularly perturbed differential difference equations (SPDDEs) deteriorates and it forms boundary layer [3]. Such type of equations has applications in the study of variational problems in control theory [4] and in modelling of neuronal variability [5]. E authors extend the method of matched asymptotic expansions initially developed for solving boundary value problems to obtain approximate solution for SPDDEs. In a series of papers [11,12,13,14], Kadalbajoo and Sharma developed uniformly convergent numerical methods using fitted mesh FDMs techniques. Notation 1. e symbol C is used to denote positive constant independent of ε and N. e norm ‖.‖ denotes the maximum norm
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