Partial differential equations simple waves

Any disturbance W [e,g, the displacement of an elastic string or the height of a water wave) which is propagated along the r-direction with velocity u satisfies a simple partial differential equation, the wave equation ... 

The aim of molecular orbital theory is to provide a complete description of the energies of electrons and nuclei in molecules. The principles of the method are simple a partial differential equation is set up, the solutions to which are the allowed energy levels of the system. However, the practice is rather different, and, just as it is impossible (at present) to obtain exact solutions to the wave equations for polyelectronic atoms, so it is not possible to obtain exact solutions for molecular species. Accordingly, the application of molecular orbital theory to molecules is in a regime of successive approximations. Numerous rigorous mathematical methods have been utilised in the effort to obtain ever more accurate solutions to the wave equations. This book is not concerned with the details of the methods which have been used, but only with their results. 

The outline of this chapter is as follows First, some basic wave phenomena for separation, as well as integrated reaction separation processes, are illustrated. Afterwards, a simple mathematical model is introduced, which applies to a large class of separation as well as integrated reaction separation processes. In the limit of reaction equilibrium the model represents a system of quasilinear first-order partial differential equations. For the prediction of wave solutions of such systems an almost complete theory exists [33, 34, 38], which is summarized in a second step. Subsequently, application of this theory to different integrated reaction separation processes is illustrated. The emphasis is placed on reactive distillation and reactive chromatography, but applications to other reaction separation processes are also... 

When a wide rectangular injection pulse is injected in a column and the width is such that the plateau is not completely eroded when it is eluted, the solution of the system of equations of the ideal model (Eqs. 8.1a and 8.1b) includes a constant state, followed by a simple wave, as shown by the theory of partial differential equations [12,13]. The importance of this result is due to the existence of a relationship between the concentrations of the two components of the binary mixture in the simple wave region. This relationship is independent of the position of the band along the column. We have discussed the properties of the hodograph transform in the case of the ideal model (Oiapter 8, Sections 8.1.2 and 8.8). In the case of the equilibrium-dispersive model (Eqs. 11.1 and 11.2), this result is no longer valid. However, the plots of Ci versus C2 are often close to the simple wave solu-... 

We analyze the mechanism of wave-shape pattern formation based on the proposed model and verify it by comparing the numerical computation and the experimental data. The model is represented by a rather simple nonlinear first-order partial differential equation of equation (2.7). It describes that the effect of the electric field to the gel is determined solely by the geometry of the equipo-tential surface and the gel surface. We rewrite the equation as follows in order to represent the geometry of the experimental setup. 

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