论文标题
按时间尺度求解非线性$ q $ factional微分方程的差异方法
A difference method for solving the nonlinear $q$-factional differential equations on time scale
论文作者
论文摘要
$ Q $ - 分离的微分方程通常描述在时间尺度设置$ t_q $上施加的物理过程。在本文中，我们首先提出了一个差异公式，用于离散时间尺度设置$ t_q $的分数$ q $^CD_Q^αx（t）$，订单$ 0 <α<1 $和比例索引$ 0 <q <q <1 $。我们建立了严格的截断误差界限，并证明此差异公式无条件稳定。然后，我们考虑解决$ q $ - 分离微分方程的初始问题的差异方法：$^cd_q^αx（t）= f（t，x（t））$在时间表集中。我们证明了差异解决方案的独特存在和稳定性，并提供了收敛分析。数值实验显示了所提出的差异方法的有效性和高精度。
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 $^cD_q^αx(t)$ on the time scale set $T_q$ with order $0<α<1$ 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: $^cD_q^αx(t)=f(t,x(t))$ 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.
