Differential-algebraic equations consistent initial conditions

Boundary value problems (BVPs) consist of systems of differential or differential-algebraic equations vhere the conditions are not concentrated in a single point (e.g., the initial point). The majority of the methods proposed to solve this important family of problems transform the original problem into a nonlinear system. 

An important issue in the boundary-layer problem, and in differential-algebraic equations generally, is the specification of consistent initial conditions. We think first of the physical problem (not in Von Mises form), since the inlet profiles of u, v, and T, as well as pressure p, must be specified. However, all the initial conditions are not independent, as they would be for a system of standard-form ordinary differential equations. So assuming that the axial velocity u and temperature T profiles are specified, the radial velocity must be required to satisfy certain constraints. 

A extremely simple example can serve to illustrate why consistent initial conditions are essential when using the method of lines to solve DAEs. Consider the problem illustrated in Fig. 7.5, which has one ordinary differential equation and one algebraic constraint. Obviously the solution for y is a straight line beginning at the origin with a slope of one. The solution for y2 is simply that > 2 = 1 always. The difficulty occurs if the initial condition is for y2 f 1, which is an inconsistent initial condition. 

At the channel inlet, the radial-velocity v profile cannot be specified independently of the axial-velocity and temperature profiles. Explain why this is so. Explain the implications concerning consistent initial conditions from the differential-algebraic-equation perspective. Develop an algorithm to determine the consistent initial v profile, given the u profile. 

Because the problem is represented by differential-algebraic equations all the initial conditions are not independent. Determine the surface composition that is consistent with the gas-phase initial condition of A = 1 and B = C = 0. 

---