This paper deals with the numerical solution of fractional integro-differential 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  applied the Haar wavelet method to solve the fractional integral equations, Momani et al.  applied the Adomian decomposition method to approximate solutions for fourth-order fractional integrodifferential equations, Rawashdeh  applied collocation method to study integro-differential equations of fractional order. Moreover, properties of these equations have been studied by several authors   . "
"There are four kinds of Chebyshev polynomials as in . 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,     also see,        and other publications as     ). 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. "
[Show abstract][Hide abstract] 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.
"Sugimoto  employed Fourier SM in fractional Burgers' equation, and later Blank  adopted a spline-based collocation method for a class of FODEs. This approach was later employed by Rawashdeh  for solving fractional integro-differential equations. Li and Xu   developed a space–time spectral method for a time-fractional diffusion equation with spectral convergence, which was based on the early work of Fix and Roop . "
[Show abstract][Hide abstract] ABSTRACT: Existing numerical methods for fractional PDEs suffer from low accuracy and inefficiency in dealing with three-dimensional problems or with long-time integrations. We develop a unified and spectrally accurate Petrov-Galerkin (PG) spectral method for a weak formulation of the general linear Fractional Partial Differential Equations (FPDEs) of the form , where , , in a ( )-dimensional space-time domain subject to Dirichlet initial and boundary conditions. We perform the stability analysis (in 1-D) and the corresponding convergence study of the scheme (in multi-D). 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 Sturm-Liouville problems (FSLPs), recently introduced in Zayernouri and Karniadakis (2013). Specifically, we employ the eigenfunctions of the FSLP of first kind (FSLP-I), called Jacobi poly-fractonomials, as temporal/spatial bases. Next, we construct a different space for test functions from poly-fractonomial eigenfunctions of the FSLP of second kind (FSLP-II). Besides the high-order spatial accuracy of the PG method, we demonstrate its efficiency and spectral accuracy in time-integration schemes for solving time-dependent 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 space-fractional advection (TSFA), parabolic FPDEs such as time- and space-fractional 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 higher-dimensions 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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