18 1. Differential Equations 1.0 0.5 0.5 1.0 0.2 0.4 0.6 0.8 1.0 1.2 Figure 1.6-2: The output of the SimpleEvolver program with initial profile u(x, 0) = x2. encounter with this procedure so that you can imagine how such a numerical solution might be computed. However, you should realize that the procedure we have seen here is extremely simplistic. Just as there are better ways to approximate the value of a function than linear approximation, there are better ways to numerically estimate solutions to differential equations. In particular, it would take a much more accurate approximation scheme to produce an accurate repre- sentation of the dynamics over a longer interval of time rather than just a single small timestep as was considered here. Chapter 1: Problems 1. For each of the equations listed, identify it as being an ordinary dif- ferential equation or a partial differential equation. Also, identify whether it is linear or nonlinear. (a) f 2 + f = cos(x), (b) ψxx + ψyy = ψxy, (c) uux = utt, (d) ex d2y dx2 + e−x dy dx = cosh(x).
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