This thesis presents two different numerical methods to solve non-linear stiff differential equations. The first method is exponential integrator, its error bounds are derived for the specific differential equations. Error analysis of exponential integrators is studied based on the Frèchet differentiation and Sobolev space. We obtain the error bounds in Hs(R) norms under the certain assumptions. The second method is a new iterative linearizaton technique. For the second one, we first time applied to general Frèchet derivative as a linearization technique for the numerical solution of nonlinear partial differential equations. In computational part, in order to denote the effectiveness of the new proposed method, we compare our proposed method with the well-known techniques with respect to the errors.