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

Abstract

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.