A study on numerical algorithms for differential equation in two cases $q$-Calculus and $(p,q)$-Calculus

We investigate the existence and uniqueness of the solution and also the rate of convergence of a numerical method for a fractional differential equation in both q-calculus and (p, q)-calculus versions. We use the Banach and Schauder fixed point theorems in this study. We provide two examples, one by definition of the q-derivative and the other by (p, q)-derivative. We compare the rate of convergence of the numerical method. We like to clear some facts on (p, q)-calculus. The data from our numerical calculations show well that q-calculus works better than (p, q)-calculus in each case.