Article
Numerical solution of fractional integrodifferential equations by collocation method
Applied Mathematics and Computation (Impact Factor: 1.55). 05/2006; 176(1):16. DOI: 10.1016/j.amc.2005.09.059
Source: DBLP
ABSTRACT
This paper deals with the numerical solution of fractional integrodifferential equations of the typeDqy(t)=p(t)y(t)+f(t)+∫0tK(t,s)y(s)ds,t∈I=[0,1]by polynomial spline functions. We derive a system of equations that characterizing the numerical solution. Some numerical examples are also provided to illustrated our results.

 "For instance we can mention the following works. Lepik [33] applied the Haar wavelet method to solve the fractional integral equations, Momani et al. [38] applied the Adomian decomposition method to approximate solutions for fourthorder fractional integrodifferential equations, Rawashdeh [43] applied collocation method to study integrodifferential equations of fractional order. Moreover, properties of these equations have been studied by several authors [1] [3] [51]. "
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ABSTRACT: The purpose of the present paper is to propose an efficient numerical method for solving the differential equations of Bratutype with fractional order in reproducing kernel Hilbert space. The exact solution is calculated in the form of a convergent series with easily computable components. Finally, some examples are given to illustrate the efficiency and applicability of the method. Copyright © 2015 John Wiley & Sons, Ltd.Mathematical Methods in the Applied Sciences 07/2015; DOI:10.1002/mma.3588 · 0.92 Impact Factor 
 "There are four kinds of Chebyshev polynomials as in [14]. The majority of books dealing with Chebyshev polynomials, contain mainly results of Chebyshev polynomials of all kinds T n ðxÞ; U n ðxÞ; V n ðxÞ and W n ðxÞ and their numerous uses in different applications, and research papers dealing with some types of these polynomials (see for instance, [1] [2] [16] [17] also see, [22] [23] [24] [25] [26] [27] [28] and other publications as [4] [7] [8] [12] [20]). However, there are only a limited researches of literature on shifted Chebyshev polynomials of the second kind U Ã n ðxÞ , either from theoretical or practical points of view it uses in various applications. "
Article: Second kind shifted Chebyshev polynomials for solving space fractional order diffusion equation
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ABSTRACT: In this paper, an efficient numerical method for solving space fractional order diffusion equation is presented. The numerical approach is based on shifted Chebyshev polynomials of the second kind where the fractional derivatives are expressed in terms of Caputo type. Space fractional order diffusion equation is reduced to a system of ordinary differential equations using the properties of shifted Chebyshev polynomials of the second kind together with Chebyshev collocation method. The finite difference method is used to solve this system of equations. Several numerical examples are provided to confirm the reliability and effectiveness of the proposed method.Chaos Solitons & Fractals 04/2015; 73. DOI:10.1016/j.chaos.2015.01.010 · 1.45 Impact Factor 
 "Sugimoto [14] employed Fourier SM in fractional Burgers' equation, and later Blank [27] adopted a splinebased collocation method for a class of FODEs. This approach was later employed by Rawashdeh [28] for solving fractional integrodifferential equations. Li and Xu [29] [30] developed a space–time spectral method for a timefractional diffusion equation with spectral convergence, which was based on the early work of Fix and Roop [31]. "
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ABSTRACT: Existing numerical methods for fractional PDEs suffer from low accuracy and inefficiency in dealing with threedimensional problems or with longtime integrations. We develop a unified and spectrally accurate PetrovGalerkin (PG) spectral method for a weak formulation of the general linear Fractional Partial Differential Equations (FPDEs) of the form , where , , in a ( )dimensional spacetime domain subject to Dirichlet initial and boundary conditions. We perform the stability analysis (in 1D) and the corresponding convergence study of the scheme (in multiD). The unified PG spectral method applies to the entire family of linear ,  and like equations. We develop the PG method based on a new spectral theory for fractional SturmLiouville problems (FSLPs), recently introduced in Zayernouri and Karniadakis (2013). Specifically, we employ the eigenfunctions of the FSLP of first kind (FSLPI), called Jacobi polyfractonomials, as temporal/spatial bases. Next, we construct a different space for test functions from polyfractonomial eigenfunctions of the FSLP of second kind (FSLPII). Besides the highorder spatial accuracy of the PG method, we demonstrate its efficiency and spectral accuracy in timeintegration schemes for solving timedependent FPDEs as well, rather than employing algebraically accurate traditional methods, especially when . Finally, we formulate a general fast linear solver based on the eigenpairs of the corresponding temporal and spatial mass matrices with respect to the stiffness matrices, which reduces the computational cost drastically. We demonstrate that this framework can reduce to hyperbolic FPDEs such as time and spacefractional advection (TSFA), parabolic FPDEs such as time and spacefractional diffusion (TSFD) model, and elliptic FPDEs such as fractional Helmholtz/Poisson equations with the same ease and cost. Several numerical tests confirm the efficiency and spectral convergence of the unified PG spectral method for the aforementioned families of FPDEs. Moreover, we demonstrate the computational efficiency of the new approach in higherdimensions e.g., (1+3), (1+5) and (1+9)dimensional problems.Computer Methods in Applied Mechanics and Engineering 01/2015; 283(1):1545–1569. DOI:10.1016/j.cma.2014.10.051 · 2.96 Impact Factor
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