Nonlinear One-Dimensional Problems
The mathematical models considered in this chapter involve only a single nonlinear differential equation with one unknown, which is one-dimensional in the space variable. These equations are encountered mostly in problems of radiation heat transfer, stress in plastics bars, non-Newtonian fluid flows between parallel plates, and turbulent flows in tubes. We introduce the standard Newton’ method, the method of steepest descent, and some nonlinear conjugate gradient methods for numerical solutions of the corresponding finite element nonlinear problems. Both Galerkin and Rayleigh-Ritz finite element methods are used to drive the finitedimensional finite element equations from nonlinear differential equations and their respective boundary conditions in idealized situations. For simplicity, only linear elements are used in the finite element methods.
KeywordsSteep Descent Couette Flow Linear Element Linear Finite Element Gray Body
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