The causal α -exponential and the solution of fractional linear time-invariant systems
The paper introduces an innovative approach to solving fractional linear systems using \(\alpha\)-exponential monomials, without the need for traditional integral transforms. This new method provides a direct solution, which simplifies the computational process and enhances efficiency and accuracy, particularly for large time values.\N\NThe key contribution of the paper is the development of a methodology that avoids the use of integral transforms, which are typically employed in solving fractional linear systems. Instead, the authors utilize \(\alpha\)-exponential monomials to provide a direct approach to solving these systems. This method is advantageous as it simplifies the computational process while maintaining a high level of accuracy. The research builds upon the foundational work of \textit{B. Shahrrava} [``Closed-form impulse responses of linear time-invariant systems: a unifying approach'', in: IEEE Signal Process. Mag. 35, No. 4, 126--132 (2018; \url{doi:10.1109/MSP.2018.2810300})] by extending the methodology to fractional cases and addressing both impulse and step responses. The approach effectively views the fractional system as a cascade connection of two systems, reducing the problem to finding the impulse response. This streamlined process is both innovative and practical.\N\NThe paper also addresses numerical accuracy and effectiveness by demonstrating that the proposed solution converges uniformly to a continuous function for positive time values. The authors employ the integral representation of the Mittag-Leffler function to overcome numerical issues associated with series solutions, which ensures the robustness of the method for large time values. This is a significant improvement over traditional methods, which often struggle with accuracy and convergence issues in such cases. Furthermore, the inclusion of practical examples and numerical simulations in the paper validates the effectiveness of the proposed method. The authors also provide MATLAB code in Appendix 2, which supports the practical application of their method in real-world scenarios, making it a valuable resource for researchers and practitioners in the field.\N\NOne of the major strengths of the paper is its innovative approach to solving fractional linear systems. By providing a direct method that avoids the complexities associated with integral transforms, the authors offer a practical solution that is both efficient and effective. The comprehensive numerical analysis presented in the paper further demonstrates the method's effectiveness in various scenarios, including cases with large time values where conventional methods might fail. This shows the method's potential to become a standard tool in the study of fractional calculus and its applications in engineering and applied mathematics.\N\NHowever, there are areas where the paper could be improved. While the authors reference previous work, a more detailed comparison with other existing methods could provide a clearer understanding of the relative advantages of the proposed approach. Additionally, including more diverse numerical examples, especially those involving non-linear or more complex systems, could further demonstrate the versatility and robustness of the method. Such additions would not only strengthen the paper's contribution but also provide a more comprehensive understanding of its applicability in various contexts.\N\NIn conclusion, the paper provides a significant contribution to the study of fractional linear systems by presenting a method that simplifies the solution process while maintaining high accuracy and computational efficiency. The direct approach using \(\alpha\)-exponential monomials offers a promising alternative to traditional methods, marking a noteworthy advancement in the field of applied mathematics and engineering. Future research could explore further applications of this method and its potential integration with other numerical techniques, enhancing its utility and impact in various scientific and engineering disciplines.
scientific article; zbMATH DE number 7885487
scientific article