In the present work, a numerical technique for solving a general form of nonlinear fractional order integro-differential equations (GNFIDEs) with linear functional arguments using Chebyshev series is presented. The recommended equation with its linear functional argument produces a general form of delay, proportional delay, and advanced non-linear arbitrary order Fredholm–Volterra integro-differential equations. Spectral collocation method is extended to study this problem as a matrix discretization scheme, where the fractional derivatives are characterized in the Caputo sense. The collocation method transforms the given equation and conditions to an algebraic nonlinear system of equations with unknown Chebyshev coefficients. Additionally, we present a general form of the operational matrix for derivatives. The introduced operational matrix of derivatives includes arbitrary order derivatives and the operational matrix of ordinary derivative as a special case. To the best of authors’ knowledge, there is no other work discussing this point. Numerical test examples are given, and the achieved results show that the recommended method is very effective and convenient.

In this article, the solution for general form of fractional order delay-differential equations (GFDDEs) is introduced. The proposed GFDDEs have multi-term of integer and fractional order derivatives for delayed or non delayed terms. An operational matrix is presented for all terms. The spectral collocation method is used to solve the proposed GFDDEs as a matrix discretization method. The reliability and efficiency of the proposed scheme are demonstrated by some numerical experiments.

In this paper, a numerical technique for solving new generalized fractional order differential equations with linear functional argument is presented. The spectral Tau method is extended to study this problem, where the derivatives are defined in the Caputo fractional sense. The proposed equation with its functional argument represents a general form of delay and advanced differential equations with fractional order derivatives. The obtained results show that the proposed method is very effective and convenient.

Abstract:In this paper, shifted Chebyshev polynomials of the third kind method is presented to solve numerically the Fredholm, Volterra-Hammerstein integral equations. The proposed method converts the equation system of linear or non-linear algebraic equations, which can be solved. Some numerical examples are included to demonstrate the validity and applicability of the proposed technique. All computations are done using Mathematica 7.Keywords: Fredholm-Hammerstein integral equations, Volterra integral equation, shifted Chebyshev polynomials of the third kind method.

No abstract