Numerical analysis of fractional partial differential equations applied to polymeric visco-elastic Euler-Bernoulli beam under quasi-static loads

Abstract

• The governing equation of visco-elastic beam based on fractional derivativeis established. • The numerical solutions of deflection, stress and strain of the polymeric beam are obtained directlyin the time domain. • Error correction is proposed to improve the accuracy of the numerical algorithm. • The bending resistance of the beam with two viscoelastic polymers is discussed. In this paper, an effective numerical algorithm based on shifted Chebyshev polynomials is proposed to solve the fractional partial differential equations applied to polymeric visco-elastic problems in the time-space domain under quasi-static loads. The governing equations using local fractional rheological models based on visco-elastic properties with fractional derivatives are established. The integer and fractional differential operator matrices of polynomials are derived according to the properties of shifted Chebyshev polynomials. The fractional order governing equation is rewritten into the form of matrix product by using the polynomial to approximate the unknown function. The collocation method is used to discretize the variables and transform the original problem into an algebraic equation system. The numerical solutions of the governing equations are obtained directly in the time-domain. In addition, an error analysis including the correction method is performed. The numerical examples have been performed to identify the sensibility of the proposed governing equations and to evaluate the efficiency and accuracy of the proposed algorithm.