An adaptive mesh method for time dependent singularly perturbed differential-difference equations

Abstract

In this paper, a time dependent singularly perturbed differential-difference convection-diffusion equation is solved numerically by using an adaptive grid method. Similar boundary value problems arise in computational neuroscience in determination of the behaviour of a neuron to random synaptic inputs. The mesh is constructed adaptively by using the concept of entorpy function. In the proposed scheme, prior information of the width and position of the layers are not required. The method is independent of perturbation parameter ε and gives us an oscillation free solution, without any user introduced parameters. Numerical examples are presented to show the accuracy and efficiency of the proposed scheme.

Keywords

• Differential-difference equations
• Singular perturbation
• Boundary layer
• Entropy
• adaptive mesh
• Central finite difference scheme
• 65L11
• 65M12
• 35K20