Please cite this article as: S. Nemati, S. Sedaghat, I. Mohammadi, A fast numerical algorithm based on the second kind Chebyshev polynomials for fractional integro-differential equations with weakly singular kernels, Journal of Computational and Applied Mathematics (2016), http://dx.
AbstractA spectral method based on operational matrices of the second kind Chebyshev polynomials (SKCPs) is employed for solving fractional integro-differential equations with weakly singular kernels. Firstly, properties of shifted SKCPs, operational matrix of fractional integration and product operational matrix are introduced and then utilized to reduce the given equation to the solution of a system of linear algebraic equations. This new approach provides a significant computational advantage by converting the given original problem to an equivalent linear Volterra integral equation of the second kind with the same initial conditions. Approximate solution is achieved by expanding the functions in terms of SKCPs and employing operational matrices. Unknown coefficients are determined by solving final system of linear equations. An estimation of the error is given. Finally, illustrative examples are included to demonstrate the high precision, fast computation and good performance of the new scheme.
In this paper, the second kind Chebyshev polynomials (SKCPs) basis is used to solve time-fractional diffusion-wave equations with damping. We present some notations and definitions of the fractional calculus and introduce some basic properties of the SKCPs. Bivariate shifted SKCPs are defined and the operational matrix of fractional integration and some other needed operational matrices are constructed. Our approach uses the properties of bivariate shifted SKCPs to transform the considered problem to a matrix equation without using any collocation points. The main characteristic of this technique is that only a small number of the basic functions is needed to obtain a satisfactory result. An estimation of the error is given in the sense of Sobolev norms. Numerical examples are given to demonstrate the efficiency and accuracy of the proposed method.
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