Numerical techniques nonlinear equation solver

Equation (4.78) is a set of nonlinear algebraic equation and may be solved using various techniques [64], Often the nonlinear differential Eq. (4.77) are solved to the steady-state condition in place of the algebraic equations using the stiff ordinary differential equation solvers described in Chapter 2 [65], See Appendix I for more information on available numerical codes. 

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 ... 

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