18 - Boussinesq Equation

Abstract

This chapter discusses the Boussinesq equation that has: (1) a second order derivative in the initial value variable, (2) second-order derivatives in the spatial (boundary value) variable with respect to the dependent variable and the square of the dependent variable, and (3) a fourth order derivative in the spatial variable. The four required boundary conditions (BCs) are homogeneous in the dependent variable (Dirichlet BCs) and the second derivative of the dependent variable. The fourth-order spatial derivative is calculated by a finite difference (FD) derived specifically for fourth derivatives and by application of stage-wise differentiation in which a FD for second derivatives is used twice. The numerical solutions are computed by the method of lines (MOL), including detailed discussion of the Matlab routines and the numerical and graphical output. This chapter presents the background to the Boussinesq equation and conditions that must be satisfied for solitary wave solutions to exist. Solutions by application of direct integration and Riccati methods are obtained that match the solution used for verification of the numerical solutions. Maple code for the Riccati based solution is presented.