Algorithm

Displacement (psychology)

Classical orthogonal polynomials

Applied mathematics

Convergence (economics)

Variable (mathematics)

Economics

Computer science

Viscoelasticity

Classical mechanics

Legendre wavelet

Associated Legendre polynomials

Wavelet transform

Thermodynamics

Discrete wavelet transform

Physics

Psychology

Mathematics

Legendre polynomials

Psychotherapist

Mathematical analysis

Artificial intelligence

Numerical analysis

Economic growth

Gegenbauer polynomials

Finance

Acceleration

Fractional calculus

Order (exchange)

Wavelet

Orthogonal polynomials

Applied Mathematical Modelling

The dynamic analysis of viscoelastic pipes conveying fluid is investigated by the variable fractional order model in this article. The nonlinear variable fractional order integral-differential equation is established by introducing the model into the governing equation. Then the Shifted Legendre Polynomials algorithm is first presented for dealing with this kind of equations. The convergence analysis and numerical example verify that the algorithm is an effective and accurate technique for addressing this type complicated equation. Numerical results for dynamic analysis of viscoelastic pipes conveying fluid show the effect of parameters on displacement, acceleration, strain and stress. It also indicates that how dynamic properties are affected by the variable fractional order and fluid velocity varying. Most of all, the proposed algorithm has enormous potentials for the problem of high precision dynamics under the variable fractional order model.

Shifted Legendre Polynomials algorithm used for the dynamic analysis of viscoelastic pipes conveying fluid with variable fractional order model