Solutions of the Boussinesq Equations through Bifurcation Method

Abstract

The article established the nonlinear theory to find the solution using a new notion of bifurcation known as attractor bifurcation. It determined the bifurcation and stability of the solutions of the Boussinesq equations as well as the onset of the Rayleigh-Benard convection. In this article we considered the theory that comprises the succeeding perspectives. The study initially deal with the problem that bifurcates from the trivial solution an attractor A_R while the Rayleigh number R intersects the first critical Rayleigh number R_C for all physically boundary conditions, despite the multiplicity of the eigenvalue R_C for the linear problem. Hereafter, secondly, the study measured the bifurcated attractor A_R as asymptotically stable. Finally, the bifurcated solutions are also structurally stable when the spatial dimension is two, and are classified as a bifurcated solution as well. Furthermore, the technical method explained here provides a means, which can be adopted for manydifferen