Modified methods for solving two classes of distributed order linear fractional differential equations

This paper introduces two methods for the numerical solution of distributed order linear fractional differential equations. The first method focuses on initial value problems (IVPs) and based on the αth Caputo fractional definition with the shifted Chebyshev operational matrix of fractional integration. By applying this method, the IVPs are converted into simple linear differential equations which can be easily handled. The other method focuses on boundary value problems (BVPs) based on Picard's method frame. This method is based on iterative formula contains an auxiliary parameter which provides a simple way to control the convergence region of solution series. Several numerical examples are used to illustrate the accuracy of the proposed methods compared to the existing methods. Also, the response of mechanical system described by such equations is studied.