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Through this article, a numerical scheme based upon the modified fractional Euler method (MFEM) is introduced to find the numerical solutions of linear and nonlinear systems of fractional differential equations (SFDEs) as well as nonlinear multi-order fractional differential equations (MOFDEs). The fractional derivatives are defined by Caputo. The proposed algorithm is very simple and provides the solutions directly without linearization, perturbations or any other assumptions. Illustrating examples with numerical comparisons between the proposed algorithm and the exact and/or fourth order Runge Kutta method (RK4) are given to reveal the efficiency and the accuracy of our algorithm.

This paper introduces a new numerical mechanism for solving multi-order fractional differential equations (MOFDEs) and systems of fractional differential equations, in which the fractional derivatives are expressed in Riemman-liouville (RL) sense. A new shifted ultraspherical (Gegenbauer) operational matrix (SGOM) of fractional integration of arbitrary order is induced. By using this matrix jointly with the Tau method, the solution of fractional differential equation (FDE) is decreased to the solution of a system of algebraic equations (AEs). Helpful problems are built-in to show the powerful and validity of the proposed technique.

A numerical treatment to a system of Caputo fractional order differential-algebraic equations (SFDAEs) is presented throughout this article. The suggested method based upon the shifted Chebyshev pesedu-spectral method (SCPSM). The shifted Chebyshev polynomials (SCPs) are handled to reduce the SFDAEs into the solution of linear/ nonlinear systems of algebraic equations. By using some tested applications, the effectiveness and the accuracy of the suggested approach are demonstrated graphically. Also numerical comparisons between the proposed technique with other numerical methods in the existing literature are held. The numerical results show that the proposed technique is computationally efficient, accurate and easy to implement.

In this article, we use the operation matrix (OM) of Riemann-Liouville fractional integral of the shifted Gegenbauer polynomials with the Lagrange multiplier method to provide efficient numerical solutions to the multi-dimensional fractional optimal control problems. The proposed technique transforms the under consideration problems into sets of nonlinear equations which are easy to solve. Numerical tests including numerical comparisons with some existing methods are introduced to demonstrate the accuracy and efficiency of the suggested technique.