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A numerical method for fractional integral with applications. (English) Zbl 1142.74390
Summary: A new numerical method for the fractional integral that only stores part history data is presented, and its discretization error is estimated. The method can be used to solve the integro-differential equation including fractional integral or fractional derivative in a long history. The difficulty of storing all history data is overcome and the error can be controlled. As application, motion equations governing the dynamical behavior of a viscoelastic Timoshenko beam with fractional derivative constitutive relation are given. The dynamical response of the beam subjected to a periodic excitation is studied by using the separation variables method. Then the new numerical method is used to solve a class of weakly singular Volterra integro-differential equations which are applied to describe the dynamical behavior of viscoelastic beams with fractional derivative constitutive relations. The analytical and numerical results are compared. It is found that they are very close.

##### MSC:
 74S30 Other numerical methods in solid mechanics (MSC2010) 74D05 Linear constitutive equations for materials with memory 74K10 Rods (beams, columns, shafts, arches, rings, etc.) 65R20 Numerical methods for integral equations 45J05 Integro-ordinary differential equations 65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
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