While solving the fractional order differential equation the requirement of the higher-order derivative is obvious therefore, this paper gives a definite expression for constructing the operational matrices of derivative which is the direct method to find the derivative of higher-order according to the requirement of the total differential equation. The proposed work expands the Chebyshev polynomial of type four up to six degrees that could help get the accuracy for the numerical solution of a given differential equation. Previously Chebyshev polynomial of the third type has been used by cutting the domain from [-1, 1] to [0, 1]. This study also generates the integrational operational matrix for solving the integral equation as well as an integrodifferential equation by using the Chebyshev polynomial of type four and expand it up to six order and generate the matrix by cutting the domain from [-1, 1] to [0, 1]. This is the first attempt to generate an integrational operational matrix that has never been highlight nor generate by any researcher. Another contribution of this paper is the generation of categorical expressions for the product of two Chebyshev vectors that will help in solving the differential equation of several kinds.

Fractional calculus is one of the evolving fields in applied sciences. Delay differential equation of non-integer order plays a vital role in epidemiology, population growth, physiology economy, medicine, chemistry, control, and electrodynamics and many mathematical modeling problems Fractional Delay differential equations usually lacks analytic solutions and some of these equations can only be solved by some numerical methods. In this review article we present a comparative study on some standard numerical methods applied to solve linear fractional order differential equations with time delay. Fractional finite difference method (FFDM), Predictor-corrector method (PCM) with new and extended versions has been discussed in this article. All above mentioned methods use the Caputo type fractional differential operator to define fractional derivatives. Solution of a real-life problem formulated by FDDEs has been discussed under these methods. Results have been presented in tabular and graphical form to analyze the efficiency and scarcity of mentioned methods. These graphical and numerical comparisons are provided to illustrate and corroborate the similarity and differences between these methods.