The objective of this research is the development of an alternative method for transient analysis of dynamical systems. The proposed method consists of the traditional single frequency analysis method to capture the steady state operation of the system and a wavelet-based transient analysis which captures the disturbance. The total solution is obtained from the superposition of the periodic steady state solution and the wavelet-based disturbance solution. This thesis focuses on the second part of the solution. We have named this approach WBTA (Wavelet-Based Transient Analysis). The WBTA method is based on equivalent wavelet companion circuits of system components, such as resistors, inductors, and capacitors. This method can be implemented using any set of orthogonal wavelets. In this thesis, an implementation with Daubechies wavelets is presented. To augment the modeling of system components, the numerical integration technique and the finite difference approximation method are utilized as a means to approximate the integral and derivative of wavelet base functions. The equivalent wavelet companion circuits are developed by applying wavelet series analysis on the integro-differential equations for each element. The procedure results in a set of algebraic equations for the entire network. The solution is in terms of the wavelet coefficients of the voltages at the nodes of the network. The actual voltages can be reconstructed via wavelet series reconstruction. The results obtained using the proposed approach are compared and verified with a numerical time-domain analysis method. The advantages of the proposed method are: (1) providing trade-off between solution accuracy and computational speed, and (2) providing possible numerical stability for stiff systems.