Surfaces of general type with $p_g=1$ and $q=0$

In this paper we construct a new family of simply connected minimal complex surfaces of general type with $p_g=1$, $q=0$, and $K^2=3, 4, 5, 6, 8$ using a $\mathbb{Q}$-Gorenstein smoothing theory. We also reconstruct minimal complex surfaces of general type with $p_g=1$, $q=0$, and $K^2=1, 2$ using the same method.