Numerical Solution of Singularly Perturbed Differential-Difference Equations with Small Shifts of Mixed Type by Differential Quadrature Method

In this paper, we have presented the Differential Quadrature Method (DQM) for finding the numerical solution of boundary-value problems for a singularly perturbed differential-difference equation of mixed type, i.e., containing both terms having a negative shift and terms having a positive shift. Such problems are associated with expected first exit time problems of the membrane potential in models for the neuron. The Differential Quadrature Method is an efficient descritization technique in solving initial and/or boundary value problems accurately using a considerably small number of grid points. To demonstrate the applicability of the method, we have solved the model examples and compared the computational results with the exact solutions. Comparisons showed that the method is capable of achieving high accuracy and efficiency.