<p>This paper presents a fast and accurate numerical algorithm for computing optimal execution boundaries of the classical optimal investment problem with transaction costs under a logarithmic utility function. By transforming the original parabolic double obstacle problem into an ordinary differential problem in Fourier space, we derive a coupled nonlinear Volterra integral equation system that determines the optimal selling and buying boundaries. Although this system lacks explicit solutions, we analyze the boundaries’ asymptotic behavior under various parameters and provide their values at termination and infinity. To solve the system efficiently, we design a numerical algorithm based on Newton iteration, which demonstrates high efficiency. This algorithm is validated through numerical experiments, which illustrate the shapes of the optimal boundaries across different scenarios.</p>