Hyperbolic equation weakly

Lax PD (1954) Weak solutions of Nonlinear Hyperbolic Equations and Their Numerical Computation. Communication on Pure and Applied Mathematics 7 159-193... 

The first order derivative in Equation (2.80) corresponds to the convection in a field problem and the examples shown in Figure 2.26 illustraTes the ina bility of the standard Galerkin method to produce meaningful results for convection-dominated equations. As described in the previous section to resolve this difficulty, in the solution of hyperbolic (convection-dominated) equations, upwind-ing or Petrov-Galerkin methods are employed. To demonstrate the application of upwinding we consider the case where only the weight function applied to the first-order derivative in the weak variational statement of the problem, represented by Equation (2.82), is modified. 

The first work on pKa determination by zone electrophoresis using paper strips was described by Waldron-Edward in 1965 (15). Also, Kiso et al. in 1968 showed the relationship between pH, mobility, and p/C, using a hyperbolic tangent function (16). Unfortunately, these methods had not been widely accepted because of the manual operation and lower reproducibility of the paper electrophoresis format. The automated capillary electrophoresis (CE) instrument allows rapid and accurate pKa determination. Beckers et al. showed that thermodynamic pATt, (pATf) and absolute ionic mobility values of several monovalent weak acids were determined accurately using effective mobility and activity at two pH points (17). Cai et al. reported pKa values of two monovalent weak bases and p-aminobenzoic acid (18). Cleveland et al. established the thermodynamic pKa determination method using nonlinear regression analysis for monovalent compounds (19). We derived the general equation and applied it to multivalent compounds (20). Until then, there were many reports on pKa determination by CE for cephalosporins (21), sulfonated azo-dyes (22), ropinirole and its impurities (23), cyto-kinins (24), and so on. 

The three-dimensional, second-order, nonlinear, elliptic partial differential equation may be simplified in the limit of weak electrolyte solutions, where the hyperbolic sine of is well approximated by 4). This yields the linearized Poisson—Boltzmann equation... 

In summary, although the weakly hyperbolic nature of Eq. (8.7) has been shown rigorously only for ID phase space (i.e. one velocity component in the KE), experience strongly suggests that the full 3D system is also weakly hyperbolic. This observation implies that the numerical schemes used to solve the moment-transport equations closed with QBMM must be able to handle local delta shocks in the moments. Qne such class of numerical schemes consists of the kinetics-based finite-volume solvers presented in Section 8.2. As a final note, we should mention that the work of Chalons et al (2012) using extended Gaussian quadrature (see Section 3.3.2) and kinetics-based finite-volume solvers to close Eq. (8.7) suggests that the system with 2A + 1 moments is fully hyperbolic and thus does not exhibit... 

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