A difference method for solving the nonlinear $q$-factional differential equations on time scale

Abstract: The $q$-fractional differential equation usually describe the physics process imposed on the time scale set $T_q$. In this paper, we first propose a difference formula for discretizing the fractional $q$-derivative $^cD_q^\alpha x(t)$ on the time scale set $T_q$ with order $0<\alpha<1$ and scale index $0<q<1$. We establish a rigours truncation error boundness and prove that this difference formula is unconditionally stable. Then, we consider the difference method for solving the initial problem of $q$-fractional differential equation: $^cD_q^\alpha x(t)=f(t,x(t))$ on the time scale set. We prove the unique existence and stability of the difference solution and give the convergence analysis. Numerical experiments show the effectiveness and high accuracy of the proposed difference method.