In this thesis, we investigate a new numerical method for the evolution equations such as the generalized regularized long wave (RLW) equation and one-dimensional Burgers’ equation by the reproducing kernel functions, abbreviated as RKFs. The RLW equation has been used to study the soliton phenomena and to model a variety of phenomena such as nonlinear transverse waves in shallow water, phonon packets in nonlinear crystal and ion-acoustic and magnetohydrodynamic waves in plasma. The Burgers’ equation plays a major role in the study of nonlinear waves. It is used as a mathematical model in turbulence problems, in the theory of shock waves and in continuous stochastic process. Analytical solutions of the RLW equation can be found for the limited initial and boundary conditions, and in many cases exact solutions of the one-dimensional Burgers’ equation involves infinite series which may converge very slowly for the small values of _ 0. Hence, finding the numerical solution of the RLW and Burgers’ equations are very useful to study the physical phenomena. By using RKFs, the numerical solutions at each discrete time step is obtained by an explicit integral expression even though the scheme is truly implicit, and, hence, the computation is fully parallel. In this approach, finding a reproducing kernel function is a crucial step in solving a given boundary value problem. The structure of this thesis is as follows: In chapter one, we review the RLW and Burgers’ equations and some definitions and theorems which will be used in the next chapters. In chapter two, four numerical schemes are formulated by means of a RKF for the RLW equation and the motion of single solitary waves with different amplitude, interaction of two and three solitary waves, development of solitary waves generated by using the maxwellian initial condition and development of an undular bore are simulated. Explicit iterative formulas for the numerical solution of Burgers’ equation are formulated by applying the Hopf-Cole transformation and a RKF in chapter three. In chapter two and three, accuracy of each scheme is tested by using discrete L2? and L1?error norms, the error estimates are given and some numerical results are presented and compared with the exact solutions. Some numerical results are also compared with the results obtained by other methods. Therefore, it can be deduced that the present method is easily implemented and very effective.