%0 Journal Article
%T Spectral-Collocation Methods for Fractional Pantograph Delay-Integrodifferential Equations
%A Yin Yang
%A Yunqing Huang
%J Advances in Mathematical Physics
%D 2013
%I Hindawi Publishing Corporation
%R 10.1155/2013/821327
%X We propose and analyze a spectral Jacobi-collocation approximation for fractional order integrodifferential equations of Volterra type with pantograph delay. The fractional derivative is described in the Caputo sense. We provide a rigorous error analysis for the collocation method, which shows that the error of approximate solution decays exponentially in norm and weighted -norm. The numerical examples are given to illustrate the theoretical results. 1. Introduction Many phenomena in engineering, physics, chemistry, and other sciences can be described very successfully by models using mathematical tools from fractional calculus, that is, the theory of derivatives and integrals of fractional noninteger order. This allows one to describe physical phenomena more accurately. Moreover, fractional calculus is applied to the model frequency dependent damping behavior of many viscoelastic materials, economics, and dynamics of interfaces between nanoparticles and substrates. Recently, several numerical methods to solve fractional differential equations (FDEs) and fractional integrodifferential equations (FIDEs) have been proposed. In this paper, we consider the general linear fractional pantograph delay-integrodifferential equations (FDIDEs) with proportional delays, with , where , £¿£¿ , and £¿£¿ are given functions and are assumed to be sufficiently smooth in the respective domains. In (1), denotes the fractional derivative of fractional order . Differential and integral equations involving derivatives of noninteger order have shown to be adequate models for various phenomena arising in damping laws, diffusion processes, models of earthquake [1], fluid-dynamics traffic model [2], mathematical physics and engineering [3], fluid and continuum mechanics [4], chemistry, acoustics, and psychology [5]. Let denote the Gamma function. For any positive integer and , the Caputo derivative is defined as follows: The Riemann-Liouville fractional integral of order is defined as we note that From (4), fractional integrodifferential equation (1) can be described as Several analytical methods have been introduced to solve FDEs including various transformation techniques [6], operational calculus methods [7], the Adomian decomposition method [8], and the iterative and series-based methods [9]. A small number of algorithms for the numerical solution of FDEs have been suggested [10], and most of them are finite difference methods, which are generally limited to low dimensions and are of limited accuracy. As we know, fractional derivatives are global (they are defined by an integral
%U http://www.hindawi.com/journals/amp/2013/821327/