Abstract
This paper presents the procedure to obtain analytical solutions of Lienard 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.
Highlights
Many dynamical phenomena can be represented by Liénard equations, such as biological, mechanical, and electrical systems
The contribution presented in this article can be considered as an extension of this work, because we present another space-temporal Liénard type model for pipelines but governed by fractional derivatives, which gives the opportunity to model unknown dynamics associated to fluid phenomena in the pipeline
6 Conclusions The Liénard equation is used in many fields of science for representing the dynamical behavior of physical systems
Summary
Many dynamical phenomena can be represented by Liénard equations, such as biological, mechanical, and electrical systems. The RiemannLiouville definition entails physically unacceptable initial conditions (fractional order initial conditions) [ ]; for the Caputo representation, the initial conditions are expressed in terms of integer-order derivatives having direct physical significance [ ], Gómez-Aguilar et al Advances in Difference Equations (2016) 2016:173 these definitions have the disadvantage that their kernel has a singularity [ ], this kernel includes memory effects and both definitions cannot accurately describe the full effect of the memory Due to this inconvenience, Caputo and Fabrizio in [ ] presented a new definition of fractional operator without a singular kernel, the Caputo-Fabrizio (CF) fractional operator; this operator possesses very interesting properties, for instance, the possibility to describe fluctuations and structures with different scales [ ]. Other applications of the CF fractional operator are given in [ – ]
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