In this paper, exact periodic wave solutions for the perturbed Boussinesq equation with power law nonlinearity are obtained for different nonlinear strengths n. When $n=1$, the periodic traveling wave solutions can be found by the definition of the Jacobian elliptic function. When $n\ge 1$, we construct a transformation to solve for the power law nonlinearity, and the periodic traveling wave solutions can be obtained by applying the extended trial equation method. In addition, we consider the limiting case where the periodicity of the periodic traveling wave solutions vanishes, and we obtain the soliton solution for $n=1$. Numerical simulations show the periodicity of the solution for the perturbed Boussinesq equation.

A low-dimensional non-linear observer is developed in this paper for the estimation of the thermal distribution in a rapid thermal processing (RTP) process. Based on the reduced model obtained from Galerkin's method, a low-dimensional non-linear Luenberger observer is designed to re-construct the thermal distribution from the output data of the point temperature sensors. Test results in a RTP system show fast convergence speed and small estimated errors of the proposed observer.