In this paper, a novel type of polynomial is defined which is equipped with an auxiliary parameter a. These polynomials are a combination of the Chebyshev polynomials of the second kind. The approximate solution of each equation is assumed as the sum of these polynomials and then, with the help of the collocation points, the unknown coefficients of each polynomial, as well as auxiliary parameter, is obtained optimally. Now, by placing the optimal value of a in polynomials, the polynomials are obtained without auxiliary parameter, which is the restarted step of the present method. The time discretization is performed on fractional partial differential equations by L1 method. In the following, the convergence theorem of the method is proved.
Abbasbandy, Saeid and Hajishafieiha, Jalal
(R1491) Numerical Solution of the Time-space Fractional Diffusion Equation with Caputo Derivative in Time by a-polynomial Method,
Applications and Applied Mathematics: An International Journal (AAM), Vol. 16,
2, Article 6.
Available at: https://digitalcommons.pvamu.edu/aam/vol16/iss2/6