In this article, on the basis of our previous research on bandwidth of non-collinear phase-matching second harmonic generation (SHG) and nonlinear Cherenkov radiation (NCR) generated on the boundary of bulk nonlinear crystal, we demonstrated theoretically the coupled wave equation of SHG generated by a Gaussian beam on the boundary of 5%/mol MgO:LiNbO3. Using the Fourier-transform technique, we solved this coupled wave equation. The results can explain theoretically the generation mechanism of two different experimental phenomena which corresponded to non-collinear phase-matching SHG under anomalous-dispersion-like dispersion condition and nonlinear NCR under normal dispersion condition, respectively. The theoretical results show a good agreement with the experimental data. This work can provide theoretical fundamental for more complicated research on the boundary of nonlinear media.