1.6. Evolution in Time 15 1.6.2 Numerical Evolution We will call a differential equation for an unknown function u(x, t) an evolution equation if it happens to be written in the form ut = p(u, ux,uxx,..., ∂nu ∂x ) where p(x0,...,xn) is a polynomial in n + 1 variables. Examples of such equations include ut = uux and ut = u2 + uxxx. In this section, we will learn a simple method for determining the approximate dynamics of a solution6 to these equations over a very short time interval starting from any given initial profile u(x, 0) = u0(x). If we think of the differential equation as a rule for determining how things can evolve in time, this gives us a way to see what would happen at later times if we know the shape of the wave at time t = 0 without having to be able to find a formula for such a solution. Imagine the dynamics of each point on the curve separately. That is, consider the graph y = u0(x) in the xy-plane as being the initial state of this evolving curve and focus your attention on one point (x0, u(x0)). Whether that point will move up or down depends on the value of ut(x0, 0) if this number is positive, then it will move up and if it is negative it will move down in a tiny interval of time after t = 0. Moreover, the magnitude of this time derivative will tell how quickly it is moving up or down. Doing the same for every point on the graph provides us with the information needed to get a sense of how the graph will evolve. But, note that we can compute the exact value of ut(x0, 0) since the right-hand side of the evolution equation gives it in terms of things we know: the initial profile and its x-derivatives! Example 1.7 Consider the evolution equation ut = u(u ux) and the initial profile u(x, 0) = x2. What will happen to the points (0, 0), (−1, 1) and (1, 1) under the evolution determined by the equa- tion in a tiny interval of time after time t = 0. 6 In certain contexts, the Cauchy-Kovalevskaya theorem can guarantee us that the solution constructed in this way is not simply a solution, but the unique solution u(x, t) to the evolution equation satisfying u(x, 0) = u0(x).
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