In this paper, we study the boundary value problem on the unit circle for the Bratu's equation depending on the real parameter $µ$. From the parameter estimate, the existence of non-negative solution is set. A numerical method is suggested to justify the theoretical result. It is a combination of the adaptation of finite difference and Gauss-Seidel method allowing us to obtain a good approximation of $\mu_c$, with respect to the exact theoretical method $\mu_c=\lambda=5.7831859629467$.