The aim of this article is to obtain numerical solution for incompressible unsteady flow for third grade fluid induced by variable accelerated plate. Numerical solution is obtained by using Hybrid method which combine between finite difference method (FDM) and asymptotic interpolation method. The influence of difference values of material constant parameters on the velocity flow fluid are discussed and shown graphically.

In this work, we derive the operational matrix using poly-Bernoulli polynomials. These polynomials generalize the Bernoulli polynomials using a generating function involving a polylogarithm function. We first show some new properties for these poly-Bernoulli polynomials; then we derive new operational matrix based on poly-Bernoulli polynomials for the Atangana–Baleanu derivative. A delay operational matrix based on poly-Bernoulli polynomials is derived. The error bound of this new method is shown. We applied this poly-Bernoulli operational matrix for solving fractional delay differential equations with variable coefficients. The numerical examples show that this method is easy to use and yet able to give accurate results.

In this work, we apply the operational matrix based on shifted Legendre polynomials for solving Prabhakar fractional differential equations. The Prabhakar derivative is defined in three-parameter Mittag-Leffler function. We achieve this by first deriving the analytical expression for Prabhakar derivative of x p where p is positive integer, via integration. Hence, for the first time, the operational matrix method for Prabhakar derivative is derived by using the properties of shifted Legendre polynomials. Hence, we transform the Prabhakar fractional differential equations into a system of algebraic equations. By solving the system of algebraic equations, we were able to obtain the numerical solution of fractional differential equations defined in Prabhakar derivative. Only a few terms of shifted Legendre polynomials are needed for achieving the accurate solution.