**4. Conclusions**

This paper introduced two independent definitions for defining the fractional Laplacian of the half-order (−<sup>Δ</sup>) 12 in the distribution, both explicitly and implicitly. We demonstrate several examples, such as (−<sup>Δ</sup>) 12 *<sup>δ</sup>*(*m*)(*x*) and (−<sup>Δ</sup>) 12 arctan *x*, some of which are undefined in the classical sense. The results obtained have potential applications in solving the differential equations involving the half-order Laplacian operator. For example, the differential equation:

$$(-\Delta)^{\frac{1}{2}}u(x) = \frac{x}{1+x^2}$$

has a solution:

$$u(x) = \arctan x + ax + b$$

on any non-empty subset of *R*, and the differential equation:

$$(-\Delta)^{\frac{1}{2}}u(x) = \text{P.V.} \frac{1}{x}$$

has a solution:

$$u(\mathbf{x}) = \pi \theta(\mathbf{x}) + a\mathbf{x} + b$$

where *a* and *b* are arbitrary constants. **Author Contributions:** The order of the author list reflects contributions to the paper.

**Funding:** This work is partially supported by NSERC (Canada 2017-00001) and NSFC (China 11671251).

**Acknowledgments:** The authors are grateful to the reviewers and editor for the careful reading of the paper with several productive suggestions and corrections, which certainly improved its quality.

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