Application of Fractional Calculus for Dynamic Problems of Solid Mechanics: Novel Trends and Recent Results
01 Jan 2010-Applied Mechanics Reviews (American Society of Mechanical Engineers)-Vol. 63, Iss: 1, pp 010801
TL;DR: In this article, the authors present the analysis of new trends and recent results carried out during the last 10 years in the field of fractional calculus application to dynamic problems of solid mechanics.
Abstract: The present state-of-the-art article is devoted to the analysis of new trends and recent results carried out during the last 10 years in the field of fractional calculus application to dynamic problems of solid mechanics. This review involves the papers dealing with study of dynamic behavior of linear and nonlinear 1DOF systems, systems with two and more DOFs, as well as linear and nonlinear systems with an infinite number of degrees of freedom: vibrations of rods, beams, plates, shells, suspension combined systems, and multilayered systems. Impact response of viscoelastic rods and plates is considered as well. The results obtained in the field are critically estimated in the light of the present view of the place and role of the fractional calculus in engineering problems and practice. This articles reviews 337 papers and involves 27 figures. DOI: 10.1115/1.4000563
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TL;DR: With the numerical least square method, it is shown that the fractional model perfectly fits the test data of memory phenomena in different disciplines, not only in mechanics, but also in biology and psychology.
Abstract: Fractional derivative has a history as long as that of classical calculus, but it is much less popular than it should be. What is the physical meaning of fractional derivative? This is still an open problem. In modeling various memory phenomena, we observe that a memory process usually consists of two stages. One is short with permanent retention and the other is governed by a simple model of fractional derivative. With the numerical least square method, we show that the fractional model perfectly fits the test data of memory phenomena in different disciplines, not only in mechanics, but also in biology and psychology. Based on this model, we find that a physical meaning of the fractional order is an index of memory.
316 citations
TL;DR: In this article, a new fractional derivative of the Caputo type is proposed and some basic properties are studied, such as adaptively changing the memory length, the new definition is capable of capturing local memory effect in a distinct way, which is critical in modelling complex systems where the short memory properties has to be considered.
Abstract: In this paper, a new fractional derivative of the Caputo type is proposed and some basic properties are studied. The form of the definition shows that the new derivative is the natural extension of the Caputo one, and that it yields the Caputo derivative with designated memory length. By adaptively changing the memory length, the new definition is capable of capturing local memory effect in a distinct way, which is critical in modelling complex systems where the short memory properties has to be considered. Another attractive property of the new derivative is that it is naturally associated with the Riemann–Liouville definition and as a result, the well established Grunwald–Letnikov approach for numerically solving the fractional differential equation can be readily embedded to approximate the solution of differential equation that involves the new derivatives. Numerical simulations demonstrate the changeable memory effect of the new definition.
160 citations
TL;DR: This paper presents a computational technique based on the collocation method and Muntz polynomials for the solution of fractional differential equations with superior accuracy and exponential convergence.
Abstract: This paper presents a computational technique based on the collocation method and Muntz polynomials for the solution of fractional differential equations. An appropriate representation of the solution via the Muntz polynomials reduces its numerical treatment to the solution of a system of algebraic equations. The main advantage of the present method is its superior accuracy and exponential convergence. Consequently, one can obtain good results even by using a small number of collocation points. The accuracy and performance of the proposed method are examined by means of some numerical experiments.
150 citations
TL;DR: In this article, the spectral approximations are used to compute the fractional integral and the Caputo derivative, and effective recursive formulae based on the Legendre, Chebyshev and Jacobi polynomials are developed to approximate the integral.
Abstract: In this paper, the spectral approximations are used to compute the fractional integral and the Caputo derivative. The effective recursive formulae based on the Legendre, Chebyshev and Jacobi polynomials are developed to approximate the fractional integral. And the succinct scheme for approximating the Caputo derivative is also derived. The collocation method is proposed to solve the fractional initial value problems and boundary value problems. Numerical examples are also provided to illustrate the effectiveness of the derived methods.
139 citations
TL;DR: In this article, the pseudo-spectral method is generalized for solving fractional differential equations with initial conditions, and an appropriate representation of the solution is presented and the pseudospectral differentiation matrix of fractional order is derived.
129 citations
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TL;DR: This website becomes a very available place to look for countless perturbation methods sources and sources about the books from countries in the world are provided.
Abstract: Following your need to always fulfil the inspiration to obtain everybody is now simple. Connecting to the internet is one of the short cuts to do. There are so many sources that offer and connect us to other world condition. As one of the products to see in internet, this website becomes a very available place to look for countless perturbation methods sources. Yeah, sources about the books from countries in the world are provided.
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