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In this paper, we consider a time‐dependent singularly perturbed differential‐difference equation with small shifts arising in the field of neuroscience. The terms containing the delay and advance parameters are approximated by using the Taylor's series expansion. The continuous problem is semi‐discretized using the Crank–Nicolson finite difference method in the time direction on uniform mesh and quadratic B‐spline collocation method in the space direction on exponentially graded mesh. The method is shown to be second‐order uniformly convergent in space and time direction. Theoretical estimates are carried out which support the obtained numerical experiments. [ABSTRACT FROM AUTHOR]

This work focuses on the numerical solution of two‐dimensional singularly perturbed convection–diffusion equations via a new stable finite difference (NSFD) scheme on a tensor product of two piecewise‐uniform Shishkin meshes. First, we convert the two‐dimensional equation into two one‐dimensional equations using the alternating direction implicit technique. A NSFD scheme has been developed using Taylor's series and the one‐dimensional equations. Here the truncation of Taylor's series is different from the classical finite difference scheme. The convergence analysis is also studied on a tensor product of two piecewise‐uniform Shishkin meshes. Numerical simulations confirm the theory. Moreover, from the numerical illustrations, it is also observed that the method is parameter uniform convergent. [ABSTRACT FROM AUTHOR]

Summary: In this paper, an adaptive mesh selection strategy is presented for solving a weakly coupled system of singularly perturbed delay differential equations of convection‐diffusion type using second order central finite difference scheme. Layer adaptive mesh is generated via an entropy production operator. The details of the location and width of the boundary layer is not required in the proposed method unlike the popular layer adaptive meshes mainly by Bakhvalov and Shishkin. The method is independent of perturbation parameter and gives us an oscillation‐free solution. The applicability of the proposed method is illustrated by means of three examples. [ABSTRACT FROM AUTHOR]