Numerical analysis of axially non-linear viscoelastic string with the variable fractional order model by using Bernstein polynomials algorithm

This paper presents a new numerical method to solve the non-linear constitutive equations of viscoelastic string with variable fractional order model. Firstly, the non-linear constitutive equation of the string is established based on the variable fractional viscoelastic model. Secondly, non-linear constitutive equation of the string is transformed into a matrix equation by using Bernstein polynomials, which can greatly simplify the solving process. Then the matrix equation is discretized and solved to obtain the numerical solutions. Finally, numerical results of viscoelastic string show the effect of the axial tension and axial velocity on its displacement, velocity and acceleration. It also indicates that how dynamic behaviours are affected by external force excitation. In addition, the proposed algorithm shows enormous potentials for solving the problem of high precision dynamics with the variable fractional order model.