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In this paper, a kinetic equation of Euler–Bernoulli beam is established with variable order fractional viscoelastic model. An effective numerical algorithm is proposed. This method uses a combination of shifted Bernstein polynomial and Legendre polynomial to approximate the numerical solution. The effectiveness of the algorithm is tested and verified by mathematical examples. The dynamic behavior of viscoelastic beams made of two materials under various loading conditions is studied. [ABSTRACT FROM AUTHOR]

An effective numerical algorithm is presented to analyze the fractional viscoelastic plate in the time domain for the first time in this paper. The viscoelastic behavior of the plate is described with fractional Kelvin–Voigt (FKV) constitutive model in three-dimensional space. A governing equation with three independent variables is established. Ternary unknown function in the governing equation is solved by deriving integer and fractional order differential operational matrices of the shifted Legendre polynomials. Error analysis and mathematical example are presented to verify the effectiveness and accuracy of proposed algorithm. Finally, numerical analysis of the plate under different loading conditions is carried out. Effects of the damping coefficient on vibration amplitude of the viscoelastic plate are studied. The results obtained are consistent with the current reference and actual situation. It shows that shifted Legendre polynomials algorithm is suitable for numerical analysis of fractional viscoelastic plates. • The fractional order governing equation of a viscoelastic plate is established. • Shifted Legendre polynomials algorithm is used to solve the governing equation. • The feasibility and efficiency of the proposed algorithm are verified. • Transverse displacements of viscoelastic plate are calculated directly in the time domain. [ABSTRACT FROM AUTHOR]

This paper presents several numerical techniques to generate solitary-wave profiles of the Benjamin equation. The formulation and implementation of the methods focus on some specific points of the problem: on the one hand, the approximation of the nonlocal term is accomplished by Fourier techniques, which determine the spatial discretization used in the experiments. On the other hand, in the numerical continuation procedure suggested by the derivation of the model and already discussed in the literature, several algorithms for solving the nonlinear systems are described and implemented: the Petviashvili method, the Preconditioned Conjugate Gradient Newton method and two Squared-Operator methods. A comparative study of these algorithms is made in the case of the Benjamin equation; Newton's method combined with Preconditioned Conjugate Gradient techniques, emerges as the most efficient. The resulting numerical profiles are shown to have a high accuracy as travelling-wave solutions when they are used as initial conditions in a time-stepping procedure for the Benjamin equation. The paper also explores the generation of multi-pulse solitary waves. [ABSTRACT FROM AUTHOR]

Numerical simulations of high strain rate plastic flow have historically been built in a hypoelastic framework and use radial return (Wilkins’ method) as the solution algorithm. We show how each of these choices can lead to inaccurate and possibly nonconvergent results. We describe an alternative solution procedure based on a simple multiple time scale perturbation theory that is stable, accurate, computationally efficient and simple to implement. Further extension of these results then leads to a strain space formulation that has additional computational advantages. We illustrate our development with numerical experiments. This paper is dedicated to my friend and colleague Christo Christov on the occasion of his 60th birthday, in recognition of his many important and creative contributions to the formulation of continuum mechanics. [ABSTRACT FROM AUTHOR]

The predictor–corrector (P–C) method applies linear interpolation technique to calculate Volterra integral equations equivalent to the considered fractional differential equations (FDEs). This paper reveals that, the relative error approaches a certain value but not infinitesimal even as the step size decreases to zero for certain FDEs. In these equations, the integrated function has a zero value and an infinite (or infinitesimal) slope at the origin. The interpolation technique is responsible for the non-diminishing relative error. Based on this analysis, we modify the P–C method by employing an alternative interpolation strategy to reduce the relative error. Numerical examples show the modified method can provide much more accurate approximations not only near the origin but also over the whole solution domain. [ABSTRACT FROM AUTHOR]

The paper describes different approaches to generalize the trapezoidal method to fractional differential equations. We analyze the main theoretical properties and we discuss computational aspects to implement efficient algorithms. Numerical experiments are provided to illustrate potential and limitations of the different methods under investigation. [ABSTRACT FROM AUTHOR]