A numerical solution method for the fractional moment problem within engineering

Purpose In the field of engineering, the fractional moments of random variables play a crucial role and are widely utilized. They are applied in various areas such as structural reliability assessment and analysis, studying the response characteristics of random vibration systems and optimizing signal processing and control systems. This study focuses on calculating the fractional moments of positive random variables encountered in engineering. This study focuses on calculating the fractional moments of positive random variables encountered in engineering. Design/methodology/approach By integrating Laplace transforms with fractional derivatives, both analytical and practical numerical solutions are derived. Furthermore, specific practical application methods are provided. Findings This approach allows for the stable and highly accurate calculation of fractional moments based on the integer moments of random variables. Data experiments included in this study demonstrate the effectiveness of this method in solving fractional moment calculations in engineering. Compared to traditional methods, the proposed method offers significant advantages in stability and accuracy, which can further advance research in the engineering field that employs fractional moments. Originality/value (1) Accuracy: Although the proposed method does involve some error, its error level is significantly lower than traditional methods, such as the Taylor expansion method. (2) Stability: The computational error of the proposed method is not only minimal but also remains stable within a narrow range as the fractional order varies. (3) Efficiency: Compared to the widely used Taylor expansion method, the proposed method requires only a minimal number of integer-order moments to achieve the desired results. Additionally, it avoids convergence issues during computation, greatly reducing computational resource requirements. (4) Simplicity: The application steps of the proposed method are very straightforward, offering significant advantages in practical applications.