Direct and inverse problems in the generalized logarithmic Boussinesq equation

Boussinesq equations are used as mathematical models for investigating the behaviour of small-amplitude long waves on water surfaces. In this paper, we focus on the logarithmic Boussinesq equation. We first generalize the model to consider different model coefficients. Secondly, we provide analytical and numerical solutions to the direct problem. Thirdly, we applied the modulating function method to solve coefficients inverse problems in the generalized logarithmic Boussinesq equation. Finally, numerical simulations are given to show the stability, accuracy, and effectiveness of the proposed methods for solving the posed direct and inverse problems.