Numerical solution for a class of variable order fractional integral-differential equation with Atangana-Baleanu-Caputo fractional derivative

In this paper we consider fractional integral-differential equations of variable order containing Atangana-Baleanu-Caputo fractional derivatives as follows: begin{align*} mathfrak{D}_{alpha(t)}^{ABC}&Big[u(x,t).g(x,t)Big]+frac{partial u(x,t)}{partial t}+int_{0}^{t}u(x,Y)dYnonumber\ &+int_{0}^{t}u(x,Y).k(x,Y)dY=f(x,t), end{align*} We try to solve this equation using a numerical method based on matrix operators including Chebyshev polynomials. By using these operational matrixes the fractional order integral-differential equation is transformed into an algebraic system which by solving them, we will obtain the numerical answer of the above fractional integral-differential equation. To show the accuracy and efficiency of this method, we have calculated some numerical examples by MATLAB software.