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Chebyshev spectral method based on operational matrix is applied to both systems of fractional integro-differential equations and Abel’s integral equations. Some test problems, for which the exact solution is known, are considered. Numerical results with comparisons are made to confirm the reliability of the method. Chebyshev spectral method may be considered as alternative and efficient technique for finding the approximation of system of fractional integro-differential equations and Abel’s integral equations.

SUMMARYIn this paper, the method of deriving the Bäcklund transformation from the Riccati form of inverse method is presented for the perturbed nonlinear Schrödinger equation (PNSE). Consequently, the exact solutions for the PNSE can be obtained by the AKNS class. The technique developed relies on the construction of the wave functions that are solutions of the associated AKNS, that is, a linear eigenvalues problem in the form of a system of partial differential equation. Moreover, we construct a new soliton solution from the old one and its wave function.

This article is devoted to use the basic Lie symmetry method to obtain the admitted Lie group of the reduction of Time fractional generalized Burgers equation. One-dimensional and two-dimensional optimal system are determined for symmetry algebras obtained through classification of their subalgebras.Some invariant solutions are also found.

We study in this paper the Q‐symmetry and conditional Q‐symmetries of Drinfel'd–Sokolov–Wilson equations. The solutions which we obtain in this paper take the form of convergent power series with easily computable components. Copyright © 2014 John Wiley & Sons, Ltd.