The construction of quantum time of arrival operator conjugate to the system Hamiltonian leads to a particular linear homogeneous Goursat problem. In this work, we demonstrate how to approximate the solution of the mentioned differential equation both semi-analytically and numerically with the goal of calculating the largest eigenvalue of the associated confined time of arrival operator. In the analytical approximation, we used the partial sum expansion of the solution and showed that the resulting largest eigenvalue converges as the number of terms increases. The result shows that for the parameters considered in this paper, the approximation is sufficient up to the fourth order correction term. In the numerical approximation, we develop a non-iterative formula to obtain the numerical solution of the Goursat problem. The performance of the non-iterative method is compared with the known numerical techniques in literature. Numerical results show that the non-iterative algorithm is more accurate and faster compared to the other techniques considered in this paper. Specifically, the proposed algorithm was able to approximate the largest eigenvalue of the confined time of arrival operator up to the third correction term for a particular nonlinear system.