Integral equation method nonlinear boundary conditions

For the given scheme of partition of the machining zone boundary on the elements, the discretization of the boundary integral equation and the boundary conditions is performed. The set of nonlinear equations, which is obtained by discretization, is solved by Newton s method. As a result, the distribution of the current density over the WP surface is obtained (Fig. 10c). 

Using the boundary conditions (equations (5.54) and (5.55)) the boundary values uo and un+i can be eliminated. Hence, the method of lines technique reduces the nonlinear parabolic PDE (equation (5.48)) to a nonlinear system of N coupled first order ODEs (equation (5.52)). This nonlinear system of ODEs is integrated numerically in time using Maple s numerical ODE solver (Runge-Kutta, Gear, and Rosenbrock for stiff ODEs see chapter 2.2.5). The procedure for using Maple to solve nonlinear parabolic partial differential equations with linear boundary conditions can be summarized as follows ... 

These two equations, along with the equations for the boundary conditions, form a system of nonlinear differential equations that are very difficult to solve in closed form recourse is therefore made to computer techniques (numerical methods). With certain simplifications, direct integration is possible. The appropriate boundary conditions are found in this case by considering the balance of mass at the point M where entering fresh medium meets with the recycled medium (see Fig. 3.38). For z = 0, one then has... 

While nonlinear in g2 (the coefficients 7,.r and b are lengthy integral expressions), the partial differential equation is linear in the derivatives. It can thus be solved by the method of characteristics, with the boundary conditions given by the coupling at /x = 0, as obtained from finite-T lattice QCD. 

This method, powerful as it is, leads to a nonlinear implicit functional equation for the boundary motion, which must be solved by numerical means. In many cases a direct numerical attack on the governing equation and boundary conditions has been preferred but Kolodner s method has the advantage of being an exact integral formulation which does not require solution of the heat equation throughout all space at each step of the boundary motion. 

In general, the problem just defined is nonlinear, in spite of the fact that the governing, creeping-flow equations are linear. This is because the drop shape is unknown and dependent on the pressure and stresses, which in turn, depend on the flow. Thus n and F are also unknown functions of the flow field, and the boundary conditions (2-112), (2-122), (2-141), and (8-58) are therefore nonlinear. Thus, for arbitrary Ca, for which the deformation may be quite significant, the problem can be solved only numerically. Later in this chapter, we briefly discuss a method, known as the boundary Integral method, that may be used to carry out such numerical calculations. Here, however, we consider the limiting case... 

---