Electrical circuit analysis is a fundamental aspect of engineering, requiring accurate and efficient computational methods for solving differential equations governing circuit behaviour. Traditional numerical methods, such as Euler's and Runge-Kutta approaches, have limitations in accuracy and computational efficiency. This paper explores an autonomous numerical approach using the Taylor Series Method, implemented in the TKSL simulation system. The study compares the Taylor Series Method with conventional numerical techniques, evaluating accuracy, computational complexity, and stability. The results indicate that the Taylor expansion method enhances precision while reducing computational overhead. This work contributes to the development of efficient circuit simulation tools, with potential applications in power systems, embedded electronics, and real-time circuit analysis. Future studies will focus on extending this approach to nonlinear circuit systems and optimizing computational performance.