Why are nonlinear equations important?
Most of us were introduced to mathematics by solving linear equations. We obtained analytical solutions that can be represented by well-known functions, such as polynomials, exponential functions, sines or cosines (which have to be tabulated), Bessel functions, error functions, and so forth. In engineering studies, the linear problems provide great insight into some phenomena, but only up to a point. Consider the following linear problem. It is possible to solve this equation analytically. The norm of the solution is plotted in the figure. Note that the curves are all parallel to each other, and only displaced up or down depending on the initial condition. This is a result of the fact that it is a linear problem. Analytic solution to linear problem, eps = 10-7, 10-6, 10-5 Next consider the more general problem. where is the norm of the solution. If the norm of the solution is small enough, the problem is the same as the linear problem shown above. Indeed, the solution is identical, as
