SetDirectory["c:\\myDirectory"]

<< DDESpecialSolutions.m

DDESpecialSolutions[{D[u[n, t] , 
            t]  ==  (1 + alpha*u[n, t])*(v[n, t] - 
                v[n - 1, t]),  
        D[v[n, t],  t]  ==  
          v[n, t]*(u[n + 1, t] - u[n, t] + alpha*v[n + 1, t] - 
                alpha*v[n - 1, t])},  {u[n, t], v[n, t]}, {n, 
        t}, {alpha}]

DDESpecialSolutions[{Derivative[0, 1][u][n, t]==(1 + alpha u[n, t]) (-v[-1 + n, t] + 
                    v[n, t]),Derivative[0, 1][v][n, t]==v[n, 
                  t] (-u[n, t] + u[1 + n, t] - 
                    alpha v[-1 + n, t] + alpha v[1 + n, t])},{u[n, t], v[n, t]},{n, t},{alpha}]

DDESpecialSolutions[{D[u[n, t],  t]  ==  
          u[n, t]*(v[n, t] - v[n - 1, t]),  
        D[v[n, t],  t]  ==  
          v[n, t]*(u[n + 1, t] - u[n, t])},  {u[n, t],  v[n, t]},  {n, 
        t},  {}]

DDESpecialSolutions[{D[u[n, t], t] - 
            alpha*(u[n + 1, t] - 2*u[n, t] + u[n - 1, t]) - 
            u[n, t]*v[n, t]*(u[n + 1, t] + u[n - 1, t]) == 0,  
        D[v[n, t], t] + alpha*(v[n + 1, t] - 2*v[n, t] + v[n - 1, t]) + 
            u[n, t]*v[n, t]*(v[n + 1, t] + v[n - 1, t]) == 0},  {u[
          n, t],  v[n, t]},  {n, t},  {alpha}]