The numerical solutions of the time-independent Schrödinger equation (TISE) for different one-dimensional potentials forms are sometime achieved by the asymptotic iteration method. However, in practice, the main difficulty in using this method is related to the choice of the adjustable parameter, which is responsible to determine the rate of convergence. In this work, the eigenvalues of energies and their corresponding wave functions for ten different anharmonic oscillator systems with potential form: ( ) ∑ with 0 Nb  have been done by using novel numerical methods which are depend on the power series method. Then, the new results have been matched by comparing them with the previous ones which have been calculated by the numerical methods. g Finally, the adjustable parameter for each system that leads to accurate results has been discussed and determined. In this thesis, the chapters are arranged into five chapters, namely: Chapter one has previous studies about the anharmonic oscillator and its applications. Chapter two presents the traditional approximation and numerical methods that have been used to solve the Schrödinger equation. Chapter three discusses new analytical methods for solving the Schrödinger equation and their algebraic derivation expression that the programming has been based on. Chapter four contains the applications of the novel methods by using ten potentials and a comparison between our results with the previous ones. Chapter five presents a brief summary and some concluding remarks