Finally, it is intended that the results of this work clarify several theoretical aspects related to the analytical solutions of TTHEs-HL and that the analogies presented can improve the understanding of the relationship between TTHEs-HL and MBHEs-HL. where 𝑋 = 𝑥 𝐿 ⁄ is the dimensionless position. The exponential coefficients (expressed below) and the roots (𝜆′s ) are defined as Peigné et al. :