Abstarct: One of the fundamental problems in numerical analysis is solving a system of nonlinear equations. For solving most of engineering problems, we faced with a system of nonlinear equations. thus, we must always look for methods for solving these problems. One of the most widely used methods for solving a system of non-linear equations is the Newton–Raphson method. In this thesis, we will propose some iterative methods and mult-istep methods for solving a system of nonlinear equations. This methods is mainly baxsed on the Newton–Raphson method, that is used to show how to convert a common-repetitive method to a multistep method. We will proved the order of convergence of these methods and then we will refer to discussion about index performance. we will show that for solving Partial differential equations and differential equations with ordinary derivatives, first we have to use some methods of discretization, such as Chebyshev’s pseudo-spectral method to convert it to Algebraic form and then we will solve it with repetitive multi-step methods. With solving some examples, we will show that the repetitive multi-step methods have higher accuracy and order of convergence than repetitive-common methods.