Numerical analysis of viscoelastic pipe with variable fractional order model based on shifted Legendre polynomials algorithm

In this paper, an valid numerical algorithm is presented to solve variable fractional viscoelastic pipes conveying pulsating fluid in the time domain and analyze dynamically the vortex-induced vibration of the pipes. Firstly, Coimbra variable fractional derivative operators are introduced. Meanwhile, using Hamilton’s principle and a nonlinear variable fractional order model, the governing system of equations is established. The unknown functions of the system of equations are approximated with shifted Legendre polynomials. Then, convergence analysis and numerical example investigate the effectiveness and accuracy of the proposed algorithm. Finally, the influences of different parameters on the dynamic response of the viscoelastic pipe are studied. The influencing factors and their ranges of the transient and long-term chaotic states of the pipe are analyzed. In addition, the proposed algorithm shows enormous potentials for solving the dynamics problems of viscoelastic pipes with the variable fractional order models.

The optimal displacement control for fractional incommensurate mass-spring oscillators by Shifted Legendre polynomials

Based on the Riemann–Liouville fractional derivative, an optimal displacement control strategy for fractional incommensurate mass-spring oscillators has been presented. According to the calculus of variations and the Lagrange multiplier technique, the optimality conditions for the given problem are obtained. Using Shifted Legendre polynomials, the problem of solving differential equations is transformed into the problem of solving a set of algebraic equations. The validity, high efficiency and applicability of the proposed method are demonstrated through the simulation results.