Problems 1 show that the solution representation 714

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Unformatted text preview: xample 7.1.1. Find the stability constant. Show that this problem is stable and has exponential dichotomy. 8 y Yj (x) x a=x 0 x1 x j-1 xj xN Figure 7.2.1: Geometry and nomenclature for multiple shooting. 7.2 Multiple Shooting As seen in Section 7.1, errors may accumulate as the solution is generated from the initial to the terminal point. Since round-o errors increase in proportion to the number of operations, they can be reduced by integrating over shorter distances. With this procedure, called parallel or multiple shooting, we create IVPs that start at several points on a b]. Let us illustrate the procedure with a linear m-dimensional BVP of the form y = Ay + b a<x<b 0 (7.2.1a) Ly(a) = l (7.2.1b) Ry(b) = r (7.2.1c) with L of dimension l m and R of dimension r m. Let us further divide a b] into N subintervals as shown in Figure 7.2.1. Multiple shooting procedures that are based on the strategies (7.1.2 - 7.1.3) and (7.1.7 - 7.1.10) have been developed. The rst approach is the si...
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