Finding Discontinuous Solutions to the Differential-Difference Equations by the Homotopy Analysis Method

An analytic method, namely the homotopy analysis method, is applied to nonlinear problems with discontinuity governed by the differential-difference equation. Purely analytic solutions are given for nonlinear problems with discontinuity with a global convergence. This method provides a new analytical approach to solve nonlinear problems with discontinuity. Comparisons are made between the results of the proposed method and the exact solutions. The results reveal that the proposed method is very effective and convenient.