In recent decades, differential equations and their numerical solutions have been used extensively in the natural sciences, engineering, and financial mathematics. Differential equations and developing analytical and numerical methods for the solutions of fractional differential equations with variableorder fractional derivatives have essential applications in the parts of biomathematics, mathematics, chemistry, electronics, economics, engineering, etc. In the last years and also, study and discussion this type of differential equation for some modeling problems of the differential equations containing Riemann-Liouville, and Caputo derivative definitions are the most valuable tools in fractional calculus [5, 7, 10, 11, 18, 21, 25]. Recently, the numerical schemes for the solution of a class of variable-order fractional differential equations (FDEs) and the solutions of FDEs with fractional derivative have beenstudied, for example variational iteration method [16, 34], Adomian decomposition method [3, 6], generalized differential transform method [12], Wavelet Method [2], finite difference method [24, 34], collocation method [26], Expression of numerical, methods with the help of Chebyshev polynomials (CPs) [27] andcubic spline interpolation method [13] and other methods [5, 8, 14] must be used. We will discuss in this paper is given replacing Riemann-Liouville fractional integrals with Atangana-Baleanu integrals in the definition of Riemann-Liouville derivatives.

Variable-order Fractional, Atangana-Baleanu-Caputo Fractional Derivative, Chebyshev Polynomials.