unconditionalstability

The fractional calculus deals with the generalization of integration and differentiation of integer order to those ones of any order .The q-fractional differential equation usually describes the physical process imposed on the time scale set Tq. In this paper, we first propose a difference formula for discretizing the fractional q-derivative 𝐷 𝑐 𝑞 𝛼𝑥(𝑡), of Caputo type with order 0 < 𝛼 < 1 and scale index 0 < 𝑞 < 1. We establish a rigorous 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: 𝐷𝑞 𝛼𝑥(𝑡) = 𝑓(𝑡, 𝑥(𝑡)) 𝑐 . We prove the 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.

In this scheme, we first present a difference formula (called the 𝐿1,𝑞 formula) to discretize the fractional q-derivative 𝐷 𝑐 𝑞 𝛼𝑥(𝑡), of Caputo type with 0 < 𝛼, 𝑞 < 1. This difference formula is constructed by using the piecewise linear interpolation to approximate the integral function. Then, by using this difference formula, we establish a difference method for solving the initial problem of q-fractional differential equation.

We prove that this difference method is unconditionally stable and give an error estimation of ∆𝑡𝑛 2 -order. Numerical experiments show the high accuracy and effectiveness of this difference formula. To the authors’ best knowledge, it is the first time that an unconditionally stable difference formula is presented and analyzed for the q-fractional problems. Our work provides a numerical approach for solving the q-fractional problems.

The following conclusions were drawn from this research.  We present the difference formula and drive truncation error boundness.  The formula is contributed to the stability analysis.  The difference technique is used to solve the initial value problem of q-fractional differential equation.  The existence of the solution, stability and error estimation are given for the difference formula.