The partial differential equation (PDE) solution of the telegrapher is a promising fault location method among time-domain and model-based techniques. Recent research works have shown that the leap-frog process is superior to other explicit methods for the PDE solution. However, its implementation is challenged by determining the initial conditions in time and the boundary conditions in space. This letter proposes two implicit solution methods for determining the initial conditions and an analytical way to obtain the boundary conditions founded on the signal decomposition. The results show that the proposal gives fault location accuracy superior to the existing leap-frog scheme, particularly in the presence of harmonics.