**4. Conclusions**

In this paper, we have obtained the unique existence result of approximate solution of initial value problem for fractional differential equation of variable order involving with the variable order derivative defined on the half-axis. Through discussing the characters of variable order calculus(integral and derivative), we introduce the concept of approximate solution to the problem. Based on our discussion and analysis, using the fixed point theorem, we have found the unique existence results. As applications, two examples are presented to illustrate the main results. The issue of the existence and qualitative analysis of approximate solution of initial value problems for fractional differential equation of variable order is interesting. In the future, we will consider the existence and qualitative analysis of approximate solution of initial value problem for singular fractional differential equation of variable order.

**Author Contributions:** All authors contributed equally in writing this article. All authors read and approved the final manuscript.

**Funding:** This research is funded by the National Natural Science Foundation of China (11671181). This research is also supported by Doctor Science Foundation of Shandong Jiaotong University and Shandong Jiaotong University "Climbing" Research Innovation Team Program.

**Conflicts of Interest:** The authors declare no conflict of interest.
