This paper presents the procedure to obtain analytical solutions of Liénard type model of a fluid transmission line represented by the Caputo-Fabrizio fractional operator. For such a model, we derive a new approximated analytical solution by using the Laplace homotopy analysis method. Both the efficiency and the accuracy of the method are verified by comparing the obtained solutions with the exact analytical solution. Good agreement between them is confirmed.

The purpose of this paper is to provide a structural review of the progress made on the detection and localization of leaks in pipelines by using approaches based on the Kalman filter. To the best of the author’s knowledge, this is the first review on the topic. In particular, it is the first to try to draw the attention of the leak detection community to the important contributions that use the Kalman filter as the core of a computational pipeline monitoring system. Without being exhaustive, the paper gathers the results from different research groups such that these are presented in a unified fashion. For this reason, a classification of the current approaches based on the Kalman filter is proposed. For each of the existing approaches within this classification, the basic concepts, theoretical results, and relations with the other procedures are discussed in detail. The review starts with a short summary of essential ideas about state observers. Then, a brief history of the use of the Kalman filter for diagnosing leaks is described by mentioning the most outstanding approaches. At last, brief discussions of some emerging research problems, such as the leak detection in pipelines transporting heavy oils; the main challenges; and some open issues are addressed.