Approximate Solution of Nonlinear Boussinesq Equation

This study presents an approximate solution of the nonlinear Boussinesq equation (NBE). The approximate solution of NBE is assumed to consist of space function and the Fourier cosine series which has time-varying coefficients. The space function is very thin and constant in time. The summation of the space function and the Fourier cosine series satisfies the initial and boundary conditions. When the bottom is horizontal or sloped, the coefficients in the Fourier cosine series are analytically determined by Galerkin’s method (one of the methods of weighted residuals) and the consequent nonlinear ordinary differential equations are numerically solved by the fourth order Runge-Kutta-Gill method. The resultant numerical results are found to be in satisfactory agreement with the experimental data. When the effect of the bottom slope in NBE is assumed to be linearly distributed in the computational space domain, an approximate solution is easily found to give the fluid surface profile on the sloped bottom as the fluid surface profile on the horizontal cases. The computational result by the finite difference method is also compared with the experimental data and the approximate solution.