## 1 Introduction  
In this paper, we consider an algorithm for the localization and classification of faults in a signal based on wavelet analysis and numerical methods: WONC-FD (Wavelet-Based Optimization and Numerical Computing for Fault Detection). Fault detection is an important problem in seismic signal processing [1, 2, 3, 4], vibration analysis in industrial equipment [5, 6, 7, 8, 9], and structural and electrical network monitoring [10, 11, 12]. This approach uses wavelet transform to localize faults in the signal for subsequent classification using optimization techniques through minimization of the target error function. This study focuses on the development of the WONC-FD (Wavelet-Based Optimization and Numerical Computing for Fault Detection) algorithm for the accurate detection and categorization of faults in signals using wavelet analysis augmented with numerical methods. Fault detection is a key problem in areas related to seismic activity analysis, vibration assessment of industrial equipment, structural integrity control, and electrical grid reliability. In the proposed methodology, wavelet transform serves to accurately localize anomalies in the data, and optimization techniques are introduced to refine the classification based on minimizing the error function. This not only improves the accuracy of fault identification but also provides a better understanding of their nature. The development of an algorithm based on wavelet analysis and additional mathematical methods will be an important step in the development of signal processing theory and fault diagnosis methods. This approach will increase the accuracy and sensitivity of fault detection, which are critical for ensuring the reliability and safety of electrical systems. The solution of the designated scientific problem has practical and theoretical potential. Effective fault diagnosis allows one to promptly identify and eliminate problems, minimizing the risk of accidents and equipment damage. The developed algorithm can be adapted for use in various industries where the stable operation of electrical devices is important, such as industry, energy, transportation, and household appliances. This will minimize risks and ensure the more reliable operation of electrical systems, which is an urgent task in modern industry. Modern requirements for the quality of the maintenance and operation of electrical systems entail more accurate and reliable diagnostic methods, and the developed algorithm fully meets these requirements, contributing to their satisfaction. The current state of research on fault diagnosis in electrical circuits demonstrates significant progress in the development of signal processing and machine learning techniques. However, existing methods often face limitations in accuracy and sensitivity, especially when detecting complex and immediate faults. Traditional methods such as Fourier transform do not provide detailed time-frequency analysis of signals, making it difficult to detect minor changes in signals. In this regard, the search for new and more effective diagnostic methods becomes one of the key tasks of modern science and engineering. One of the most promising directions in the field of signal processing is the use of wavelet transform [13; 14; 15; 16; 17]. Wavelet analysis allows for detailed time-frequency analysis of signals, which makes it particularly attractive for diagnosing failures in electrical circuits. Many studies on the application of wavelet transform in various fields, including medical diagnosis, audio signal processing, and data analysis, confirm its effectiveness and flexibility. In particular, wavelet transform can detect small and complex changes in signals, which is critical for fault diagnosis in electrical circuits. Examples of the use of wavelet transform in fault diagnosis are in the works [18; 19; 20; 21], where the authors apply wavelet analysis to detect faults in electrical networks. Wavelet transform can identify anomalies in voltage and current signals, which improves the diagnostic accuracy.