International Journal of Mathematics and Mathematical Sciences
Volume 22 (1999), Issue 3, Pages 643-648
doi:10.1155/S0161171299226439
Existence of periodic traveling wave solutions to the generalized forced Boussinesq equation
Department of Mathematics and Computer Science, Fayetteville State University, Fayetteville 28301-4298, North Carolin, USA
Received 3 September 1998
Copyright © 1999 Kenneth L. Jones and Yunkai Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
The generalized forced Boussinesq equation, utt−uxx+[f(u)]xx+uxxxx=h0, and its periodic traveling wave solutions are considered. Using the transform
z=x−ωt, the equation is converted to a nonlinear ordinary differential equation with periodic boundary conditions. An equivalent relation between the ordinary differential equation and a Hammerstein type integral equation is then established by using the Green's function method. This integral equation generates compact operators in a Banach space of real-valued continuous
periodic functions with a given period 2T. The Schauder's fixed point theorem is then used to prove the existence of solutions to the integral equation. Therefore, the existence of nonconstant
periodic traveling wave solutions to the generalized forced Boussinesq equation is established.